Lecture Notes on Algebraic Topology II John Rognes November 22nd 2011

Contents 1 Singular homology and cohomology 1.1 Chain complexes . . . . . . . . . . . . . . . . . 1.2 Some homological algebra . . . . . . . . . . . . 1.3 Singular homology . . . . . . . . . . . . . . . . 1.4 Tensor product and Hom-groups . . . . . . . . 1.5 Homology with coefficients . . . . . . . . . . . . 1.6 Relative homology . . . . . . . . . . . . . . . . 1.7 The Eilenberg–Steenrod axioms for homology . 1.8 Singular cohomology . . . . . . . . . . . . . . . 1.9 Relative cohomology . . . . . . . . . . . . . . . 1.10 The Eilenberg–Steenrod axioms for cohomology 1.11 Cellular homology and cohomology . . . . . . .

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universal coefficient theorems Half-exactness . . . . . . . . . . . . . . . . . . . . Free resolutions . . . . . . . . . . . . . . . . . . . Tor and Ext . . . . . . . . . . . . . . . . . . . . . The universal coefficient theorem in homology . . The universal coefficient theorem in cohomology Some calculations . . . . . . . . . . . . . . . . . . Field coefficients . . . . . . . . . . . . . . . . . .

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25 26 27 28 29 31 33 35

3 Cup product 3.1 The Alexander–Whitney diagonal approximation 3.2 The cochain cup product . . . . . . . . . . . . . . 3.3 The cohomology cup product . . . . . . . . . . . 3.4 Relative cup products, naturality . . . . . . . . . 3.5 Cross product . . . . . . . . . . . . . . . . . . . . 3.6 Relative cross products, naturality . . . . . . . . 3.7 Projective spaces . . . . . . . . . . . . . . . . . . 3.8 Hopf maps . . . . . . . . . . . . . . . . . . . . . . 3.9 Graded commutativity . . . . . . . . . . . . . . . 3.10 Tensor products of graded rings . . . . . . . . . .

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37 37 40 40 43 44 45 49 51 52 53

2 The 2.1 2.2 2.3 2.4 2.5 2.6 2.7

4 K¨ unneth theorems 54 4.1 A K¨ unneth formula in cohomology . . . . . . . . . . . . . . . . . 54 4.2 The K¨ unneth formula in homology . . . . . . . . . . . . . . . . . 54 4.3 Proof of the cohomology K¨ unneth formula . . . . . . . . . . . . . 56 i

CONTENTS 5 Poincar´ e duality 5.1 Orientations . . . . . . . . . . . . . . 5.2 Cap product . . . . . . . . . . . . . . 5.3 Cohomology with compact supports 5.4 Duality for noncompact manifolds . 5.5 Connection with cup product . . . . 5.6 Other forms of duality . . . . . . . .

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61 61 65 65 66 68 71

6 Vector bundles and classifying spaces 72 6.1 Real vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 72 6.2 Other kinds of vector bundles . . . . . . . . . . . . . . . . . . . . 75 6.3 Constructing new bundles out of old . . . . . . . . . . . . . . . . 77 6.4 Grassmann manifolds and universal bundles . . . . . . . . . . . . 80 6.5 Oriented bundles and the Euler class . . . . . . . . . . . . . . . . 81 6.6 The Thom isomorphism theorem . . . . . . . . . . . . . . . . . . 81 6.7 Chern classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.8 Pontryagin classes . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.9 Bordism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.9.1 Thom complexes and the Thom isomorphism . . . . . . . 81 6.9.2 Tubular neighborhoods and the Pontryagin–Thom construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.9.3 Transversality and Thom’s theorem . . . . . . . . . . . . 82 6.9.4 Homology of Thom spectra . . . . . . . . . . . . . . . . . 83 6.9.5 Pontryagin numbers . . . . . . . . . . . . . . . . . . . . . 83 6.9.6 Computations in low dimensions . . . . . . . . . . . . . . 84 6.9.7 The index formula . . . . . . . . . . . . . . . . . . . . . . 84

Foreword These are notes intended for the author’s Algebraic Topology II lectures at the University of Oslo in the fall term of 2011. The main references for the course will be: • Allen Hatcher’s book “Algebraic Topology” [2], drawing on chapter 3 on cohomology and chapter 4 on homotopy theory. • John Milnor and Jim Stasheff’s book “Characteristic Classes” [3]. • Pierre Conner’s book “Differentiable Periodic Maps” [1]. Comments and corrections are welcome—please write to [email protected] .

iii

Introduction Cohomology (Co-)homology theories There are various ways of associating to each topological object, like a topological space or a differentiable manifold, an algebraic object, like a group or a graded commutative ring. This can be interesting because of what the resulting algebraic object tells us about the topological object, or because known topological examples can produce novel algebraic examples. This is often the general framework of algebraic topology. Usually the algebraic objects are constructed by comparing the given topological object, say a topological space X, with familiar topological objects, like the standard simplices ∆n or the complex plane/line C, or specially designed topological spaces, like the Eilenberg–Mac Lane spaces K(G, n). For example, to study singular homology, one considers the continuous maps σ : ∆n → X for all n ≥ 0, assembles these into the singular chain complex (C∗ (X), ∂), and passes to homology, to obtain the singular homology groups Hn (X) for n ≥ 0. This is a standard approach in algebraic topology. The construction involves maps into X, and is covariant in X, in the sense that for a map f : X → Y there is an induced homomorphism f∗ : Hn (X) → Hn (Y ) in the same direction. As another example, one may consider the commutative ring C(X) of continuous maps ϕ : X → C under pointwise addition and multiplication. These are the global sections in a sheaf of rings that to each open subset U ⊆ X associates the ring C(U ) of continuous functions on U . Under suitable assumptions on X one may consider refined versions of this: if X is a complex variety one can consider the ring O(X) of holomorphic maps ϕ : X → C. Using sheaf cohomology one can associate cohomology groups H n (X) to these ringed spaces. This is a standard approach in algebraic geometry. The construction involves maps out of X, and is contravariant in X, in the sense that for a (regular) map f : X → Y there is an induced homomorphism f ∗ : H n (Y ) → H n (X) in the opposite direction. If X is a smooth (infinitely differentiable) manifold one can consider the ring C ∞ (X) of smooth maps ϕ : X → R. Each point p ∈ X determines a maximal ideal mp , and the rule ϕ 7→ dϕ(p) induces an isomorphism mp /m2p ∼ = Tp∗ X to the cotangent space at p, dual to the tangent space. Gluing these vector spaces together one can view each differential n-form ω on X as a section in a vector bundle over X. These can be assembled into the de Rham complex (Ω∗ X, d),

1

CONTENTS

2

∗ whose cohomology defines the de Rham cohomology HdR (X). This is a standard approach in differential topology. The differential forms on X are again maps out of X, and the construction is contravariant in X. There is a variant of singular homology, called singular cohomology, which is also contravariant. Its construction is of somewhat mixed variance, since it is given in terms of functions out of things given by maps into X. More precisely, one considers functions ϕ : {n-simplices in X} → G from the set of singular n-simplices σ : ∆n → X to a fixed abelian group G. This is equivalent to considering homomorphisms ϕ : Cn (X) → G from the free abelian group of singular n-chains on X. From these functions or homomorphisms one forms a cochain complex C ∗ (X; G), whose cohomology groups are the singular cohomology groups H n (X; G). And then there is a more directly contravariant construction, valid for all topological spaces X. For each abelian group G and each n ≥ 0 there exists a topological space K(G, n), well-defined up to homotopy equivalence, such that the group πi K(G, n) = [S i , K(G, n)] of homotopy classes of maps S i → K(G, n) is trivial for i 6= n, and is identified with G for i = n. Such a space is called an Eilenberg–Mac Lane complex of type (G, n). The group [X, K(G, n)] of homotopy classes of maps X → K(G, n) defines a cohomology theory in X, which is isomorphic to the singular cohomology group H n (X; G) for a large class of spaces X. Given this wealth of possible constructions, the good news is that there are interesting uniqueness theorems: For large classes of reasonable topological spaces the various constructions agree. The formulation and proof of these theorems is best done in the language of category theory, in terms of functors and natural transformations, which was originally developed by Eilenberg and Mac Lane, largely for this purpose. The result is in some sense surprising, since it is not so clear that an abelian group built out of the continuous maps ∆n → X should have much to do with another abelian group built out of the continuous maps X → C or X → K(G, n). Consider for example the space X = Q of rational numbers, with the subspace topology from R. Any continuous map ∆n → Q is constant, so to the eyes of singular homology and cohomology, Q could equally well have had the discrete topology. On the other hand, not every map Q → C is continuous, so to the eyes of sheaf cohomology, the choice of topology on Q makes an essential difference. The standard techniques of singular (co-)homology, like homotopy invariance, the long exact sequence of a pair, excision, behavior on sums, and the dimension axiom, suffice to prove uniqueness results for the homology and cohomology of CW spaces, i.e., spaces that can be given the structure of a CW complex, and more generally for all spaces that are of the homotopy type of a CW complex. Any manifold or complex variety is a CW space, so for geometric purposes, this class of spaces is usually fully adequate for topological work. On the other hand, the space of rational numbers mentioned above is not of the homotopy type of a CW complex. When going outside of this class of spaces, there are many variant (co-)homology theories, often with special properties that may be useful in particular settings.

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Cup product The (co-)homology groups of a topological space are useful in classification of general classes of spaces, and in answering questions about special classes of spaces. The classification problem concerns questions like: “what are the possible spaces of this type?” and “given a space, which one is it?” Since the (co-)homology groups of a space are usually quite easy to compute, and two abelian groups can usually quite easily be compared to each other, it is useful to try to answer these questions in terms of the (co-)homology groups of the space. As a first step towards determining what possibilities there are for a class of topological objects, one should then determine what possibilities there are for the corresponding class of algebraic objects. Here it turns out to be fruitful to consider the (co-)homology groups as examples of a richer algebraic structure than just a sequence of abelian groups. One extra structure comes from the same source as the commutative ring structure on the set C(X) of continuous functions on a space X. This was given by the pointwise sum and product of functions, so given two maps ϕ, ψ : X → C, we can form the sum given by (ϕ + ψ)(p) = ϕ(p) + ψ(p) and the product given by (ϕ · ψ)(p) = ϕ(p)ψ(p). To make it clearer what structures are involved, we might express these formulas in terms of diagrams. Since the right hand sides in these expressions involve evaluation at p two times, we need to make two copies of that point. This is done using the diagonal map ∆: X → X × X that takes p ∈ X to (p, p) ∈ X. The sum of ϕ and ψ is then given by the composite map ϕ×ψ ∆ + X −→ X × X −→ C × C −→ C , where the last map is the addition in C, and similarly for the product. The commutativity of the product is derived from the fact that the composite ∆

τ

τ ∆ : X −→ X × X −→ X × X is equal to ∆, where τ : X × X → X × X is the twist homeomorphism that takes (p, q) to (q, p). What is the associated structure in (co-)homology? The diagonal map induces a homomorphism ∆∗ : Hn (X) −→ Hn (X × X) , but this lands in the homology of X × X, not the homology of X. With a little care it is possible to define a homology cross product map × : Hi (X) ⊗ Hj (X) −→ Hi+j (X × X) for all i, j ≥ 0, and these assemble to a map M Hi (X) ⊗ Hj (X) −→ Hn (X × X) . i+j=n

CONTENTS

4

In general this map is not an isomorphism. If it were, we could have composed ∆∗ with the inverse isomorphism, and obtained a homomorphism M Hn (X) −→ Hi (X) ⊗ Hj (X) i+j=n

for all n. As a convention, the tensor product of two graded abelian groups is defined so that the collection of all of these maps could be written as H∗ (X) −→ H∗ (X) ⊗ H∗ (X) , which would make the homology groups H∗ (X) into a “graded coring”. The combined cross product map is an isomorphism if we work with homology with coefficients in a field, and this is one reason to consider homology groups with coefficients. However, the algebraic structures of corings or coalgebras are unfamiliar ones. It is therefore most often more convenient to dualize, and to consider cohomology instead of homology. Let R be a commutative ring, for instance the ring of integers Z. The diagonal map induces a homomorphism ∆∗ : H ∗ (X × X; R) −→ H ∗ (X; R) and there is a cohomology cross product map × : H i (X; R) ⊗ H j (X; R) −→ H i+j (X × X; R) for all i, j ≥ 0. The composite is a homomorphism ∪ = × ◦ ∆∗ : H i (X; R) × H j (X; R) → H i+j (X; R) for all i, j ≥ 0, called the cup product. We may assemble these cup product maps to a pairing ∪ : H ∗ (X; R) ⊗ H ∗ (X; R) −→ H ∗ (X; R) , which makes H ∗ (X; R) a graded ring, or more precisely, a graded R-algebra. In fact, the cohomology cross product map, taking a⊗b to a×b, is compatible with the twist homeomorphism τ , in the graded sense that τ ∗ (a × b) = (−1)ij b × a, where i and j are the degrees of a and b, respectively, so that the cohomology ring H ∗ (X; R) becomes graded commutative. In much work in algebraic topology, it is therefore standard to consider the cohomology H ∗ (X; R) of a space X, not as a graded abelian group, but as a graded commutative ring or algebra. This enriched algebraic structure is still manageable, but often carries much more useful information than the plain group structure.

Poincar´ e duality Much geometric work is concerned with manifolds, or smooth varieties, rather than general topological spaces. In an n-dimensional manifold there is a certain duality between k-dimensional subobjects and suitable (n − k)-dimensional subobjects. For example, each compact, convex polyhedron in R3 determines a cell structure on its boundary, a topological 2-sphere, dividing it into vertices

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(0-cells), edges (1-cells) and faces (2-cells). There is also a dual cell structure, with a 0-cell for each of the old faces, a 1-cell for each of the old edges, and a 2-cell for each of the old vertices. We can superimpose these cell structures, so that each of the old k-cells meets one of the new (2 − k)-cells, in a single point. Algebraically, this is reflected in a certain duality in the homology, or cohomology, of a manifold. It says that for suitable n-manifolds X (closed, connected and oriented) there is a preferred isomorphism H n (X; R) ∼ = R, and the cup product pairing ∪ H k (X; R) ⊗ H n−k (X; R) −→ H n (X; R) ∼ =R

defines a perfect pairing modulo torsion. This is the Poincar´e duality theorem. If R = F is a field this means that the corresponding homomorphisms H k (X; F ) −→ Hom(H n−k (X; F ), F ) = H n−k (X; F )∗ are isomorphisms, for all k. This homomorphism takes a ∈ H k (X; F ) to the homomorphism H n−k (X; F ) → F that takes b to the image of a ∪ b ∈ H n (X; F ) in F , under the preferred isomorphism. In particular dimF H k (X; F ) = dimF H n−k (X; F ) for all k. This kind of symmetry, between dimension k and codimension k phenomena in the (co-)homology of an n-manifold, is the key feature taken as the starting point for the classification of manifolds, as a special class of topological objects among all topological spaces.

Characteristic classes ((More later))

Bordism ((More later))

Chapter 1

Singular homology and cohomology Following Hatcher’s book “Algebraic Topology” [2], we first review the definition of singular homology, and then introduce singular homology with coefficients and singular cohomology.

1.1

Chain complexes

A chain complex is a diagram ∂

∂

· · · → Cn+1 −→ Cn −→ Cn−1 → . . . of abelian groups (or R-modules, or objects in a more general abelian category), such that the composite ∂ 2 = ∂∂ : Cn+1 −→ Cn−1 is the zero homomorphism, for each integer n. We also use the abbreviated notation (C∗ , ∂), or just C∗ , for the diagram above. The elements of Cn are called n-chains. We think of C∗ as a graded abelian group, with Cn in degree n and ∂ of degree −1. This is the standard convention in algebraic topology. Let Bn = Bn (C∗ , ∂) = im(∂ : Cn+1 → Cn ) be the group of n-boundaries, and let Zn = Zn (C∗ , ∂) = ker(∂ : Cn → Cn−1 ) . be the group of n-cycles. Then Bn ⊆ Zn ⊆ Cn since any element of Bn has the form x = ∂y, and then ∂x = ∂ 2 y = 0. In general, the inclusion Bn ⊆ Zn may be a proper inclusion. To detect the possible difference, we form the quotient group Hn (C∗ , ∂) = Zn /Bn , 6

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called the n-th homology group of (C∗ , ∂). A necessary and sufficient condition for an n-cycle x ∈ Zn to be an n-boundary is then that its equivalence class (= coset) [x] ∈ Zn /Bn = Hn (C∗ , ∂) is zero. We call the equivalence class [x] the homology class of the cycle x. If there is no difference between cycles and boundaries, so that Bn = im(∂) is equal to Zn = ker(∂), as subgroups of Cn , then we say that the chain complex is exact at Cn . This is equivalent to the vanishing Hn (C∗ , ∂) = 0 of the n-th homology group. A chain complex is exact if it is exact at each object in the diagram. An exact chain complex is also called a long exact sequence. An exact chain complex of the form j i 0 → A −→ B −→ C → 0 (extended by 0’s in both directions) is called a short exact sequence. Exactness at A means that i is injective, exactness at B means that im(i) = ker(j), and exactness at C means that j is surjective. If we identify A with its image i(A) ⊆ B, this means that j induces an isomorphism B/A ∼ = C. Let (C∗ , ∂) and (D∗ , ∂) be two chain complexes. A chain map f# : C∗ → D∗ is a commutative diagram / Cn+1

...

fn+1



/ Dn+1

...

/ Cn

∂

∂

/ ...

fn−1

fn

 / Dn

∂

/ Cn−1

∂

 / Dn−1

/ ...

of abelian groups. In other words, it is a sequence of group homomorphisms fn : Cn → Dn such that ∂fn = fn−1 ∂ : Cn → Dn−1 , for all n. A chain map f# : C∗ → D∗ restricts to homomorphisms Bn (C∗ , ∂) → Bn (D∗ , ∂) and Zn (C∗ , ∂) → Zn (D∗ , ∂), hence induces a homomorphism of quotient groups: f∗ : Hn (C∗ , ∂) → Hn (D∗ , ∂) for each n. If g# : D∗ → E∗ is another chain map, then we have the relation (gf )∗ = g∗ f∗ : Hn (C∗ , ∂) → Hn (E∗ , ∂) for each n, saying that the homology groups Hn (C∗ , ∂) are (covariant) functors of the chain complex (C∗ , ∂). (We omit to mention the identity condition.) If the groups are reindexed by superscripts: C m = C−m we obtain a diagram δ

δ

· · · → C m−1 −→ C m −→ C m+1 → . . . such that the composite δ 2 = δδ : C m−1 −→ C m+1 is the zero homomorphism, for each m. This is called a cochain complex. We abbreviate this to (C ∗ , δ), or just C ∗ . The elements of C m are called m-cochains.

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Again C ∗ is a graded abelian group, with C m in degree m and δ of degree +1. This is the standard convention in algebraic geometry. Let B m = im(δ : C m−1 → C m ) and Z m = ker(δ : C m → C m+1 ) be the groups of m-coboundaries and m-cocycles, respectively. Then Bm ⊆ Z m ⊆ C m as before, and the quotient group H m (C ∗ , δ) =

Zm Bm

is called the m-th cohomology group of (C ∗ , δ). If C ∗ is obtained from C∗ by the reindexing C m = C−m , then B m = B−m , Z m = Z−m and H m (C ∗ , δ) = H−m (C∗ , ∂).

1.2

Some homological algebra

((Short exact sequence of chain complexes, connecting homomorphism, long exact sequence in homology, naturality, chain homotopy, five-lemma.))

1.3

Singular homology

For each n ≥ 0, let the standard n-simplex ∆n be the subspace ∆n = {(t0 , t1 , . . . , tn ) ∈ Rn+1 | each ti ≥ 0,

n X

ti = 1}

i=0

of Rn+1 consisting of all convex linear combinations (t0 , t1 , . . . , tn ) =

n X

ti vi

i=0

of the (n + 1) unit vectors v0 , v1 , . . . , vn , where vi = (0, . . . , 0, 1, 0, . . . , 0) has a single 1 in the i-th position, counting from 0. We call ti the i-th barycentric coordinate of the point (t0 , t1 , . . . , tn ). We call vi the i-th vertex of ∆n . Note that ∆n has (n + 1) vertices. For each 0 ≤ i ≤ n, with n ≥ 1, there is an affine linear embedding δni : ∆n−1 → ∆n , called the i-th face map, that takes (t0 , . . . , tn−1 ) ∈ ∆n−1 to δni (t0 , . . . , tn−1 ) = (t0 , . . . , ti−1 , 0, ti , . . . , tn−1 ) . In other words, it takes the j-th vertex of ∆n−1 to the j-th vertex of ∆n for 0 ≤ j < i, and to the (j + 1)-th vertex of ∆n for i ≤ j ≤ n − 1. In this way it omits the i-th vertex of ∆n , and induces the unique order-preserving

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correspondence between the n vertices of ∆n−1 with the remaining n vertices of ∆n+1 . The image of δni is the subspace of ∆n where the i-th barycentric coordinate ti is zero: δni (∆n−1 ) = {(t0 , . . . , tn ) ∈ ∆ | ti = 0} We call this part of the boundary of ∆n the i-th face. The Pn topological boundary of ∆n , as a subspace of the hyperplane in Rn+1 where i=0 ti = 1, is the union of these faces: n [ ∂∆n = δni (∆n−1 ) . i=0

Let X be any topological space. A map (= a continuous function) σ : ∆n → X is called a singular n-simplex in X. Let the singular n-chains Cn (X) = Z{σ : ∆n → X} be the free abelian group generated by the set of singular n-simplices in X. Its elements are finite formal sums X nσ σ , σ

where σ ranges over the maps ∆n → X, each nσ is an integer, and only finitely many of the nσ are different from zero. This abelian group can also be written as the direct sum M Cn (X) = Z σ : ∆n →X

of one copy of the integers for each singular n-simplex. For each singular n-simplex σ : ∆n → X, and each face map δni : ∆n−1 → ∆n , the composite map σδni = σ ◦ δni : ∆n−1 → X is a singular (n − 1)-simplex in X. Under the identification of ∆n−1 with the i-th face in the boundary of ∆n , we can think of σδni as the restriction of σ to that subspace. We call this (n − 1)-simplex the i-th face of σ, and use one of the notations σ|[v0 , . . . , vi−1 , vi+1 , . . . , vn ] = σ|[v0 , . . . , vˆi , . . . , vn ] , where the “hat” indicates a term to be omitted. The restriction of σ to the boundary of ∆n is not itself a simplex, but ∂∆n is covered by the (n + 1) faces δni (∆n−1 ), and we define the boundary of σ as a sum of the corresponding faces σδni . For reasons having to do with the ordering of the vertices of a simplex, or more precisely, with the orientation of a simplex, it turns out to be best to make this an alternating sum, with the i-th face taken with the sign (−1)i . For each singular n-simplex σ : ∆n → X in X, with n ≥ 1, let the boundary ∂σ be the singular (n − 1)-chain ∂σ =

n X i=0

(−1)i σδni =

n X i=0

(−1)i σ|[v0 , . . . , vˆi , . . . , vn ] .

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More generally, define the boundary homomorphism ∂ : Cn (X) → Cn−1 (X) to be the additive extension of this rule, so that X X ∂( nσ σ) = nσ ∂σ . σ

σ

It is then a consequence of the relation j i δn+1 ◦ δni = δn+1 ◦ δnj−1 : ∆n−1 → ∆n+1

for 0 ≤ i < j ≤ n + 1 (both maps omit the i-th and j-th vertices), that ∂ 2 = 0 : Cn+1 (X) → Cn−1 . Hence the diagram ∂

∂

· · · → Cn+1 (X) −→ Cn (X) −→ Cn−1 (X) → · · · → C0 (X) → 0 → . . . is a chain complex, called the singular chain complex of X. By convention, Cn (X) = 0 for n < 0. We consider the group Bn (X) = im(∂ : Cn+1 (X) → Cn (X)) of singular n-boundaries, and the group Zn (X) = ker(∂ : Cn (X) → Cn−1 (X)) of singular n-cycles, both of which are subgroups of Cn (X), and call the quotient group Zn (X) Hn (X) = Bn (X) the n-th singular homology group of X.

1.4

Tensor product and Hom-groups

Let A and G be abelian groups. The tensor product A ⊗ G is the abelian group generated by symbols a ⊗ g, with a ∈ A and g ∈ G, subject to the bilinearity relations (a + a0 ) ⊗ g = a ⊗ g + a0 ⊗ g and a ⊗ (g + g 0 ) = a ⊗ g + a ⊗ g 0 for a, a0 ∈ A and g, g 0 ∈ G. The Hom-group Hom(A, G) is the abelian group of group homomorphisms f : A → G, with the group operation given by pointwise addition: (ϕ + ϕ0 )(a) = ϕ(a) + ϕ0 (a) for ϕ, ϕ0 : A → G, a ∈ A. The sum ϕ + ϕ0 is a group homomorphism since G is abelian. If f : A → B is a homomorphism of abelian groups, then there are induced homomorphisms f∗ = f ⊗ 1 : A × G → B ⊗ G

CHAPTER 1. SINGULAR HOMOLOGY AND COHOMOLOGY

11

given by f∗ (a ⊗ g) = f (a) ⊗ g for a ∈ A, g ∈ G, and f ∗ = Hom(f, 1) : Hom(B, G) → Hom(A, G) given by f ∗ (ψ)(a) = ψ(f (a)) for ψ : B → G, a ∈ A. Note how the direction of the map f ∗ is reversed, compared to that of f and f∗ . If g : B → C is a second homomorphism, then we have the relations (gf )∗ = g∗ f∗ : A ⊗ G → C ⊗ G and (gf )∗ = f ∗ g ∗ : Hom(C, G) → Hom(A, G) , saying that (−) ⊗ G is a covariant functor and Hom(−, G) is a contravariant functor (in the indicated variable). If A = Z, then there is a natural isomorphism Z ⊗ G ∼ = G, taking n ⊗ Lg to the multiple ng formed in the group G. More generally, if A = Z{S} = S Z is the free abelian group generated by a set S, then M Z{S} ⊗ G ∼ G = S

is the direct sum of one copy of G for each element of S. If A = Z, then there is a natural isomorphism Hom(Z, G) ∼ = G, taking L ϕ : Z → G to the value ϕ(1) at 1 ∈ Z. If A = Z{S} = S Z then Y Hom(Z{S}, G) ∼ G = S

is the product of one copy of G for each element of S. A homomorphism ϕ : Z{S} → G corresponds to the sequence (ϕ(s))s∈S in G, of values of ϕ at the generators s ∈ S viewed as elements of Z{S}.

1.5

Homology with coefficients

Let X be any topological space and G any abelian group. The singular chain complex of X with coefficients in G is the diagram ∂⊗1

∂⊗1

· · · → Cn+1 (X) ⊗ G −→ Cn (X) ⊗ G −→ Cn−1 (X) ⊗ G → . . . . Here (∂ ⊗ 1)(∂ ⊗ 1) = ∂ 2 ⊗ 1 = 0, by functoriality, so this is indeed a chain complex. Note that M Cn (X) ⊗ G ∼ G = σ : ∆n →X

is the direct sum of one copy of the group G for each singular n-simplex. Its elements are finite formal sums X gσ σ , σ

where σ ranges over the singular n-simplices in X, each gσ is an element of G, and only finitely many of them are nonzero.

CHAPTER 1. SINGULAR HOMOLOGY AND COHOMOLOGY

12

We also use the notations Cn (X; G) = Cn (X) ⊗ G, Bn (X; G) = im(∂ ⊗ 1 : Cn+1 (X; G) → Cn (X; G)) and Zn (X; G) = ker(∂ ⊗ 1 : Cn (X; G) → Cn−1 (X; G)) for the singular n-chains, n-boundaries and n-cycles in X with coefficients in G, respectively. We often abbreviate ∂ ⊗ 1 to ∂. By definition, the n-th singular homology group of X with coefficients in G is the quotient group Hn (X; G) =

Zn (X; G) = Hn (C∗ (X; G), ∂) . Bn (X; G)

For example, let X = ? be a single point. Then there is a unique singular n-simplex σn : ∆n → ? for each n ≥ 0, so Cn (?) = Z{σn } and Cn (?; G) = G{σn } P for each n ≥ 0. We have σn δni = σn−1 for each 0 ≤ i ≤ n, n ≥ 1, so n ∂σn = i=0 (−1)i σn−1 equals σn−1 for n ≥ 2 even, and equals 0 for n ≥ 1 odd. Hence C∗ (?; G) appears as follows: 1

0

1

0

. . . −→ G{σ3 } −→ G{σ2 } −→ G{σ1 } −→ G{σ0 } → 0 The boundary homomorphisms labeled 1 are isomorphisms and the ones labeled 0 are trivial. Hence Bn (?; G) equals G{σn } for n ≥ 1 odd, and is zero otherwise, while Zn (?; G) equals G{σn } for n ≥ 1 odd, or for n = 0, and is zero otherwise. Thus for n 6= 0 we have Bn (?; G) = Zn (?; G) and Hn (?; G) = 0. In the case n = 0 we have H0 (?; G) = Z0 (?; G)/B0 (?; G) = G{σ0 }/0 ∼ = G. Let f : X → Y be any map of topological spaces. There is an induced chain map f# = C∗ (f ; G) : C∗ (X; G) → C∗ (Y ; G) given by the formula X X f# ( gσ σ) = gσ f σ . σ

σ

n

Here σ : ∆ → X ranges over the singular n-simplices of X, and the composite σ

f

f σ : ∆n −→ X −→ Y is an n-simplex of Y . This is a chain map because the associativity of composition, (f σ)δni = f (σδni ), implies that ∂(f σ) = f (∂σ). Hence there is an induced homomorphism of homology groups, f∗ = Hn (f ; G) : Hn (X; G) → Hn (Y ; G) for all n. If g : Y → Z is a second map, then the relation (gf )∗ = g∗ f∗ : Hn (X; G) → Hn (Z; G) holds.

CHAPTER 1. SINGULAR HOMOLOGY AND COHOMOLOGY

1.6

13

Relative homology

Let A ⊆ X be any subspace. Write i : A → X for the inclusion map. The chain map i# : C∗ (A; G) → C∗ (X; G) is injective in each degree, identifying each simplex σ : ∆n → A with the composite iσ : ∆n → X. Let the group of relative n-chains in (X, A) with coefficients in G be the quotient group Cn (X, A; G) =

Cn (X; G) Cn (A; G)

of n-chains in X modulo the n-chains in A. Since i# is a chain map, there is an induced boundary homomorphism ∂ : Cn (X, A; G) → Cn−1 (X, A; G) given by taking the equivalence class of an n-chain x in X modulo n-chains in A to the equivalence class of the (n − 1)-chain ∂x in X modulo (n − 1)-chains in A. Since ∂ 2 = 0 in C∗ (X; G), we must have ∂ 2 = 0 in C∗ (X, A; G), so (C∗ (X, A; G), ∂) is a chain complex. We write Bn (X, A; G) = im(∂ : Cn+1 (X, A; G) → Cn (X, A; G)) and Zn (X, A; G) = ker(∂ : Cn (X, A; G) → Cn−1 (X, A; G)) like before, and define the n-th singular homology group of the pair (X, A) with coefficients in G to be the quotient group Hn (X, A; G) =

Zn (X, A; G) = Hn (C∗ (X, A; G), ∂) . Bn (X, A; G)

Let j# : Cn (X; G) → Cn (X, A; G) be the canonical quotient homomorphism. Then j# is a chain map. Drawing the chain complexes vertically and the chain maps horizontally, we have a commutative diagram .. .

.. .

∂

0

 / Cn (A; G)

∂

0

∂

∂

 / Cn (X; G)

i#

∂

 / Cn−1 (A; G)

.. . j#

∂

i#

∂

/0

∂

 / Cn−1 (X; G)

 .. .

 / Cn (X, A; G)

j#

 / Cn−1 (X, A; G) ∂

 .. .

/0

 .. .

with exact rows. We usually draw this more compactly as the following short exact sequence of chain complexes: i#

j#

0 → C∗ (A; G) −→ C∗ (X; G) −→ C∗ (X, A; G) → 0

CHAPTER 1. SINGULAR HOMOLOGY AND COHOMOLOGY

14

Note that if A = ∅ is empty, then j# is an isomorphism of chain complexes ∼ C∗ (X, ∅; G), and j∗ is an isomorphism Hn (X; G) ∼ C∗ (X; G) = = Hn (X, ∅; G) for all n, so (absolute) homology is a special case of relative homology. There is a connecting homomorphism in homology ∂ : Hn (X, A; G) → Hn−1 (A; G) defined by taking the homology class [x] of a relative n-cycle x ∈ Zn (X, A; G) to the homology class [∂ x ˜] of the unique lift to Cn−1 (A; G) of the boundary in Cn−1 (X; G) of a representative x ˜ in Cn (X; G) of x. Here j# (˜ x) = x, so the lift exists because j# (∂ x ˜) = ∂x = 0 in Cn−1 (X, A; G). It is an (n − 1)-cycle, since its boundary in Cn−2 (A; G) maps to ∂ 2 x ˜ = 0 in Cn−2 (X; G) under the injective homomorphism i# . ((Well defined, additive.)) We refer to the pair (X, A), with A a subspace of X, as a pair of spaces. Let f : (X, A) → (Y, B) be any map of pairs of spaces. This is a map f : X → Y , subject to the condition that f (A) ⊆ B, so that we have a commutative diagram A

/X

i

f0

 B

f

 /Y

i

where f 0 denotes the restriction of f . Then we have a commutative diagram of chain complexes and chain maps 0

/ C∗ (A; G)

i#

0 f#

0

 / C∗ (B; G)

/ C∗ (X; G)

j#

i#

/0

00 f#

f#

 / C∗ (Y ; G)

/ C∗ (X, A; G)

j#

 / C∗ (Y, B; G)

/0

where the left hand square is induced by the square above, and the rows are 00 short exact sequences of chain complexes. The chain map f# on the right hand side is then determined by f# by the passage to a quotient. In particular, we have an induced homomorphism f∗ = Hn (f ; G) : Hn (X, A; G) → Hn (Y, B; G) for each n. If g : (Y, B) → (Z, C) is a second map of pairs of spaces, then (gf )∗ = g∗ f∗ . Under the isomorphism C∗ (X; G) ∼ = C∗ (X, ∅; G) we can identify 00 j# : C∗ (X; G) → C∗ (X, A; G) with the chain map j# for j equal to the map of pairs (X, ∅) → (X, A) given by the identity on X. The connecting homomorphism ∂ : Hn (X, A; G) → Hn−1 (A; G) is natural, in the sense that for any map of pairs f : (X, A) → (Y, B) the diagram Hn (X, A; G)

∂

f∗0

f∗

 Hn (Y, B; G) commutes.

/ Hn−1 (A; G)

∂

 / Hn−1 (B; G)

CHAPTER 1. SINGULAR HOMOLOGY AND COHOMOLOGY

1.7

15

The Eilenberg–Steenrod axioms for homology

Theorem 1.7.1 (Eilenberg–Steenrod axioms). Let G be a fixed abelian group and let n range over all integers. We abbreviate Hn (X, ∅; G) to Hn (X; G). (Functoriality) The rule that takes a pair of spaces (X, A) to Hn (X, A; G), and a map f : (X, A) → (Y, B) to the homomorphism f∗ : Hn (X, A; G) → Hn (Y, B; G), defines a covariant functor from pairs of spaces to graded abelian groups. (Naturality) The rule that takes a pair of spaces (X, A) to the connecting homomorphism ∂ : Hn (X, A; G) → Hn−1 (A; G) is a natural transformation. (Long exact sequence) The natural diagram ∂

j∗

i

∂

i

∗ ∗ ... Hn (X; G) −→ Hn (X, A; G) −→ Hn−1 (A; G) −→ . . . −→ Hn (A; G) −→

is a long exact sequence, where i∗ is induced by the inclusion i : A → X and j∗ is induced by the inclusion j : (X, ∅) → (X, A). (Homotopy invariance) If f ' g : (X, A) → (Y, B) are homotopic as maps of pairs, then f∗ = g∗ : Hn (X, A; G) → Hn (Y, B; G). (Excision) If Z ⊆ A ⊆ X are subspaces, so that the closure of Z is contained in the interior of A, then the inclusion (X − Z, A − Z) → (X, A) induces isomorphisms ∼ = Hn (X − Z, A − Z; G) −→ Hn (X, A) . ` (Sum) If (X, A) = α (Xα , Aα ) is a disjoint union of pairs of subspaces, then the inclusion maps (Xα , Aα ) → (X, A) induce isomorphisms M

∼ =

Hn (Xα , Aα ; G) −→ Hn (X, A) .

α

(Dimension) Let ? be a one-point space. Then H0 (?; G) = G and Hn (?; G) = 0 for n ≥ 0. The sum axiom is only interesting for infinite indexing sets, since the case of finite disjoint unions follows from the long exact sequence and excision. The dimension axiom implies that the homology of an n-dimensional disc, relative to its boundary, is concentrated in degree n. Hence for (X, A) = (Dn , ∂Dn ) the dimension n can be recovered from the homology groups H∗ (X, A; G) (for G 6= 0!). Definition 1.7.2. A functor (X, A) 7→ h∗ (X, A) and natural transformation ∂ : h∗ (X, A) → h∗−1 (A) satisfying all of the Eilenberg–Steenrod axioms for homology, except the dimension axiom, is called a generalized homology theory.

1.8

Singular cohomology

Let X be any topological space and G any abelian group. The singular cochain complex of X with coefficients in G is the diagram δ

δ

· · · → Hom(Cn−1 (X), G) −→ Hom(Cn (X), G) −→ Hom(Cn+1 (X), G) → . . .

CHAPTER 1. SINGULAR HOMOLOGY AND COHOMOLOGY

16

where δ = Hom(∂, 1) is called the coboundary homomorphism. Here δ 2 = Hom(∂ 2 , 1) = 0, by contravariant functoriality, so this is indeed a cochain complex. Note that Y Hom(Cn (X), G) ∼ G = σ : ∆n →X

is the product of one copy of the group G for each singular n-simplex. Its elements are functions ϕ : {σ : ∆n → X} −→ G where σ ranges over the singular n-simplices in X, and each value ϕ(σ) lies in G. Note that for ϕ ∈ C n−1 (X; G), δϕ ∈ C n (X; G) corresponds to the function given by the alternating sum (δϕ)(σ) = ϕ(δσ) =

n X

(−1)i ϕ(σδni ) =

i=0

n X

(−1)i ϕ(σ|[v0 , . . . , vˆi , . . . , vn ]) .

i=0

We also use the notations C n (X; G) = Hom(Cn (X), G), B n (X; G) = im(δ : C n−1 (X; G) → C n (X; G)) and Z n (X; G) = ker(δ : C n (X; G) → C n+1 (X; G)) for the singular n-cochains, n-coboundaries and n-cocycles in X with coefficients in G, respectively. By definition, the n-th singular cohomology group of X with coefficients in G is the quotient group H n (X; G) =

Z n (X; G) = H n (C ∗ (X; G), δ) . B n (X; G)

For example, let X = ? be a single point. Then there is a unique singular nsimplex σn : ∆n → ? for each n ≥ 0, so Cn (?) = Z{σn } and C n (?; G) = G{ϕn } n i for each n ≥ 0, where n ) = g. We have σn δn = σn−1 for each 0 ≤ i ≤ n, Pn (gϕ )(σ i n ≥ 1, so ∂σn = i=0 (−1) σn−1 equals σn−1 for n ≥ 2 even, and equals 0 for n ≥ 1 odd. Hence δϕn equals ϕn+1 for n ≥ 1 odd, and equals 0 for n ≥ 0 even. Hence C ∗ (?; G) appears as follows: 0

1

0

0 → G{ϕ0 } −→ G{ϕ1 } −→ G{ϕ2 } −→ G{ϕ3 } → . . . The boundary homomorphisms labeled 1 are isomorphisms and the ones labeled 0 are trivial. Hence B n (?; G) equals G{ϕn } for n ≥ 2 even, and is zero otherwise, while Z n (?; G) equals G{ϕn } for n ≥ 0 even, and is zero otherwise. Thus for n 6= 0 we have B n (?; G) = Z n (?; G) and H n (?; G) = 0. In the case n = 0 we have H 0 (?; G) = Z 0 (?; G)/B 0 (?; G) = G{ϕ0 }/0 ∼ = G. Let f : X → Y be any map of topological spaces. There is an induced cochain map f # = C ∗ (f ; G) : C ∗ (Y ; G) → C ∗ (X; G) given by the formula (f # ϕ)(σ) = ϕ(f σ) .

CHAPTER 1. SINGULAR HOMOLOGY AND COHOMOLOGY

17

Here σ : ∆n → X ranges over the singular n-simplices of X, and the composite σ

f

f σ : ∆n −→ X −→ Y is an n-simplex of Y , so that ϕ(f σ) takes values in G. This is a cochain map because the associativity of composition, (f σ)δni = f (σδni ), implies that δ(f # σ) = f # (δσ). Hence there is an induced homomorphism of cohomology groups, f ∗ = H n (f ; G) : H n (Y ; G) → H n (X; G) for all n. If g : Y → Z is a second map, then the relation (gf )∗ = f ∗ g ∗ : H n (Z; G) → H n (X; G) holds, showing that H ∗ (X; G) is a contravariant functor of X.

1.9

Relative cohomology

Let A ⊆ X be any subspace. Write i : A → X for the inclusion map. The cochain map i# : C ∗ (X; G) → C ∗ (A; G) is surjective in each degree, restricting each homomorphism ϕ : Cn (X) → G on n-chains in X to the n-chains that happen to lie in A. Let the group of relative n-cochains in (X, A) with coefficients in G be the subgroup C n (X, A; G) = ker(i# : C n (X; G) → C n (A; G)) ⊆ C n (X; G) of n-cochains in X that are zero on all n-chains that lie in A. Since i# is a cochain map, there is an induced coboundary homomorphism δ : C n−1 (X, A; G) → C n (X, A; G) given by taking an (n − 1)-cochain ϕ on X that vanishes on A to the n-cochain δϕ, which vanishes on A since the boundary of a chain in A still lies in A. Since δ 2 = 0 in C ∗ (X; G), we must have δ 2 = 0 in C ∗ (X, A; G), so (C ∗ (X, A; G), δ) is a cochain complex. We write B n (X, A; G) = im(δ : C n−1 (X, A; G) → C n (X, A; G)) and Z n (X, A; G) = ker(δ : C n (X, A; G) → C n+1 (X, A; G)) like before, and define the n-th singular cohomology group of the pair (X, A) with coefficients in G to be the quotient group H n (X, A; G) =

Z n (X, A; G) = H n (C ∗ (X, A; G), δ) . B n (X, A; G)

Let j # : C n (X, A; G) → C n (X; G) be the canonical inclusion homomorphism. Then j # is a cochain map. Drawing the cochain complexes vertically

CHAPTER 1. SINGULAR HOMOLOGY AND COHOMOLOGY

18

and the cochain maps horizontally, we have a commutative diagram .. .

.. . δ

δ

 / C n−1 (X, A; G)

0

0

δ

δ

 / C n−1 (X; G)

j#

δ

 / C n (X, A; G)

.. .

i

#

i

#

δ

j

δ

 .. .

/0

δ

 / C n (X; G)

#

 / C n−1 (A; G)  / C n (A; G) δ

 .. .

/0

 .. .

with exact rows. We usually draw this more compactly as the following short exact sequence of cochain complexes: j#

i#

0 → C ∗ (X, A; G) −→ C ∗ (X; G) −→ C ∗ (A; G) → 0 Note that if A = ∅ is empty, then j # is an isomorphism of chain complexes C (X, ∅; G) ∼ = C ∗ (X; G), and j ∗ is an isomorphism H n (X, ∅; G) ∼ = H n (X; G) for all n, so (absolute) cohomology is a special case of relative cohomology. There is a connecting homomorphism in cohomology ∗

δ : H n−1 (A; G) → H n (X, A; G) associated, as usual, to the short exact sequence of cochain complexes above. It is defined by taking the cohomology class [ϕ] of an (n − 1)-cocycle ϕ ∈ Z n−1 (A; G) to the cohomology class [δ ϕ] ˜ of the unique lift to C n (X, A; G) of the coboundary in C n (X; G) of an extension ϕ˜ in C n−1 (X; G) of ϕ. Here i# (ϕ) ˜ = ϕ, so the lift exists because i# (δ ϕ) ˜ = δϕ = 0 in C n (A; G). It is an ncocycle, since its coboundary in C n+1 (X, A; G) maps to δ 2 ϕ˜ = 0 in C n+1 (X; G) under the inclusion j # . ((Well defined, additive.)) We refer to the pair (X, A), with A a subspace of X, as a pair of spaces. Let f : (X, A) → (Y, B) be any map of pairs of spaces. This is a map f : X → Y , subject to the condition that f (A) ⊆ B, so that we have a commutative diagram A

i

f0

 B

/X f

i

 /Y

where f 0 denotes the restriction of f . Then we have a commutative diagram of cochain complexes and chain maps 0

/ C ∗ (Y, B; G)

j#

f 00#

0

 / C ∗ (X, A; G)

/ C ∗ (Y ; G)

i#

j

 / C ∗ (X; G)

/0

f 0#

f#

#

/ C ∗ (B; G)

i

#

 / C ∗ (A; G)

/0

CHAPTER 1. SINGULAR HOMOLOGY AND COHOMOLOGY

19

where the right hand square is induced by the square above, and the rows are short exact sequences of chain complexes. The cochain map f 00# on the left hand side is then determined by f # by passage to subcomplexes. In particular, we have an induced homomorphism f ∗ = H n (f ; G) : H n (Y, B; G) → H n (X, A; G) for each n. If g : (Y, B) → (Z, C) is a second map of pairs of spaces, then (gf )∗ = f ∗ g ∗ . Under the isomorphism C ∗ (X, ∅; G) ∼ = C ∗ (X; G) we can identify # ∗ ∗ j : C (X, A; G) → C (X; G) with the chain map j 00# for j equal to the map of pairs (X, ∅) → (X, A) given by the identity on X. The connecting homomorphism δ : H n−1 (A; G) → H n (X, A; G) is natural, in the sense that for any map of pairs f : (X, A) → (Y, B) the diagram H n−1 (B; G)

δ

f∗

 H n−1 (A; G)

/ H n (Y, B; G) f 00∗

δ

 / H n (X, A; G)

commutes.

1.10

The Eilenberg–Steenrod axioms for cohomology

Theorem 1.10.1 (Eilenberg–Steenrod axioms). Let G be a fixed abelian group and let n range over all integers. We abbreviate H n (X, ∅; G) to H n (X; G). (Functoriality) The rule that takes a pair of spaces (X, A) to H n (X, A; G), and a map f : (X, A) → (Y, B) to the homomorphism f ∗ : H n (Y, B; G) → H n (X, A; G), defines a contravariant functor from pairs of spaces to graded abelian groups. (Naturality) The rule that takes a pair of spaces (X, A) to the connecting homomorphism δ : H n−1 (A; G) → H n (X, A; G) is a natural transformation. (Long exact sequence) The natural diagram i∗

j∗

δ

i∗

δ

. . . −→ H n−1 (A; G) −→ H n (X, A; G) −→ H n (X; G) −→ H n (A; G) −→ . . . is a long exact sequence, where i∗ is induced by the inclusion i : A → X and j ∗ is induced by the inclusion j : (X, ∅) → (X, A). (Homotopy invariance) If f ' g : (X, A) → (Y, B) are homotopic as maps of pairs, then f ∗ = g ∗ : H n (Y, B; G) → H n (X, A; G). (Excision) If Z ⊆ A ⊆ X are subspaces, so that the closure of Z is contained in the interior of A, then the inclusion (X − Z, A − Z) → (X, A) induces isomorphisms ∼ = H n (X, A; G) −→ Hn (X − Z, A − Z; G) . ` (Product) If (X, A) = α (Xα , Aα ) is a disjoint union of pairs of subspaces, then the inclusion maps (Xα , Aα ) → (X, A) induce isomorphisms Y ∼ = H n (X, A) −→ H n (Xα , Aα ; G) α

CHAPTER 1. SINGULAR HOMOLOGY AND COHOMOLOGY

20

(Dimension) Let ? be a one-point space. Then H 0 (?; G) = G and H n (?; G) = 0 for n ≥ 0. Definition 1.10.2. A functor (X, A) 7→ h∗ (X, A) and natural transformation δ : h∗−1 (A) → h∗ (X, A) satisfying all of the Eilenberg–Steenrod axioms for cohomology, except the dimension axiom, is called a generalized cohomology theory. Proof. Contravariant functoriality of the cohomology groups, naturality of the connecting homomorphism and exactness of the long exact sequence are clear from the contravariant functoriality of the short exact sequence j#

i#

0 → C ∗ (X, A; G) −→ C ∗ (X; G) −→ C ∗ (A; G) → 0 of cochain complexes, together with the standard construction of the connecting homomorphism and exactness of the long exact sequence for short exact sequences of chain complexes. Homotopy invariance for singular cohomology follows from the proof of homotopy invariance for singular homology. Let F : (X, A) × I = (X × I, A × I) → (Y, B) be a homotopy of pairs from f : (X, A) → (Y, B) to g : (X, A) → (Y, B). For each n-simplex ∆n there is a triangulation of the cylinder ∆n × I, where I = [0, 1], which gives rise to a prism operator P : Cn (X, A) → Cn+1 (Y, B) such that ∂P + P ∂ = g# − f# : Cn (X, A) → Cn (Y, B) for each n ≥ 0. Applying Hom(−, G), we get the dual prism operator P ∗ = Hom(P, 1) : C n+1 (Y, B; G) → C n (X, A; G) such that P ∗ δ + δP ∗ = g # − f # : C n (Y, B; G) → C n (X, A; G) . For each n-cocycle ϕ ∈ Z n (Y, B; G) we have δϕ = 0, so δP ∗ (ϕ) = g # (ϕ)−f # (ϕ), which implies that f # (ϕ) and g # (ϕ) are cohomologous, i.e., they represent the same cohomology class: f ∗ [ϕ] = [f # (ϕ)] = [g # (ϕ)] = g ∗ [ϕ] . Thus f ∗ = g ∗ : H n (Y, B; G) → H n (X, A; G). Excision for singular cohomology also follows from the proof of excision for singular homology. Let B = X − Z, so that Int(A) and Int(B) cover X. Let ι : C∗ (A + B) ⊆ C∗ (X) be the inclusion of the subcomplex of simplices in A or B. Using barycentric subdivision, there is a chain map ρ : C∗ (X) −→ C∗ (A + B)

CHAPTER 1. SINGULAR HOMOLOGY AND COHOMOLOGY

21

such that ρι = 1 and 1 − ιρ = ∂D + D∂ for a chain homotopy D. Applying Hom(−, G), we get dual cochain maps ι∗ : C ∗ (X; G) −→ C ∗ (A + B; G) = Hom(C∗ (A + B), G) and ρ∗ : C ∗ (A + B; G) −→ C ∗ (X; G) such that ι∗ ρ∗ = 1 and 1 − ρ∗ ι∗ = D∗ δ + δD∗ . Hence ι∗ induces an isomorphism in cohomology. By the five-lemma for the map of long exact sequences induced by the map of short exact sequences of cochain complexes 0

/ C ∗ (A + B; G)

/ C ∗ (A + B, A; G)

ι∗

ι00

0

 / C ∗ (X, A; G)

/ C ∗ (A; G)

j

#

 / C ∗ (X; G)

/0

=

i#

 / C ∗ (A; G)

/0

it follows that also the left hand map ι∗ induces an isomorphism in cohomology. There is a natural identification C ∗ (A + B, A; G) ∼ = C ∗ (B, A ∩ B; G), and the ∗ composite of the induced isomorphism H (A+B, A; G) ∼ = H ∗ (B, A∩B; G) with ∗ ι is the excision isomorphism. The product axiom is only interesting for infinite indexing sets, since the case of finite disjoint unions ` follows from the long exact sequence and excision. Since any simplex σ : ∆n → α Xα lands Lin precisely one of the Xα ’s, there is a direct sum decomposition C∗ (X, A) ∼ = α C∗ (Xα , Aα ). Applying Hom(−, G) we get a product factorization Y C ∗ (X, A; G) ∼ C ∗ (X, A; G) . = α

Q Since each coface map δ factors as the product α δα , we also get product factorizations of B ∗ (X, A; G) and Z ∗ (X, A; G), which induce the claimed product factorization of H ∗ (X, A; G). We discussed the dimension axiom for cohomology above.

1.11

Cellular homology and cohomology

If X is a CW complex, and A ⊆ X a subcomplex, then the cellular complexes ∗ (X, A; G) are smaller complexes than the singular ones, C∗CW (X, A; G) and CCW which can be used to compute the homology and cohomology groups. Let ∅ = X (−1) ⊆ X (0) ⊆ · · · ⊆ X (n−1) ⊆ X (n) ⊆ · · · ⊆ X be the skeleton filtration of X, so that there is a pushout square ` ` n n / / α ∂D αD ϕ





Φ

/ X (n) X (n−1) / S for each n ≥ 0, and X = n≥0 X (n) has the weak (colimit) topology. The ` index α runs over the set of n-cells in X, and we decompose ϕ = α ϕα and

CHAPTER 1. SINGULAR HOMOLOGY AND COHOMOLOGY

22

` Φ = α Φα , where ϕα : ∂Dn → X (n−1) is the attaching map and Φα : Dn → X (n) ⊆ X is the characteristic map of the α-th n-cell. For each n ≥ 0, let CnCW (X) = Hn (X (n) , X (n−1) ) . By excision, homotopy invariance and the sum axiom, there is an isomorphism M M CnCW (X) ∼ Hn (Dn , ∂Dn ) ∼ Z, = = α

α

where α runs over the set of n-cells in X. Let CW dn : CnCW (X) −→ Cn−1 (X) be the composite homomorphism i

∂

∗ Hn−1 (X (n−1) , X (n−2) ) . Hn (X (n) , X (n−1) ) −→ Hn−1 (X (n−1) ) −→

This equals the connecting homomorphism in the long exact sequence of the triple (X (n) , X (n−1) , X (n−2) ) ((ETC)). Then dn dn+1 = 0. We call (C∗CW (X), d) the cellular chain complex of X, and define the cellular homology groups of X to be its homology groups HnCW (X) =

ker(dn ) = Hn (C∗CW (X), d) . im(dn+1 )

If f : X → Y is a cellular map of CW complexes, so that f (X (n) ) ⊂ Y (n) for all n, we get a homomorphism f# = CnCW (f ) = Hn (f ) : Hn (X (n) , X (n−1) ) → Hn (Y (n) , Y (n−1) ) , which defines a chain map f# : C∗CW (X) → C∗CW (Y ) and an induced homomorphism f∗ : HnCW (X) → HnCW (Y ). Hence the cellular complex and cellular homology groups are covariant functors from the category of CW complexes and cellular maps. Let A ⊆ X be a subcomplex, with skeleton filtration {A(n) }n , such that A(n) is built from A(n−1) by attaching a subset of the n-cells of X, with attaching maps landing in A(n−1) ⊆ X (n−1) . The inclusion i : A → X is cellular, and identifies the cellular chain complex C∗CW (A) with a subcomplex of C∗CW (X). Let the relative cellular n-chains CnCW (X, A) =

CnCW (X) ∼ = Hn (X (n) , X (n−1) ∪ A(n) ) CnCW (A)

be the quotient group. It is the free abelian group generated by the n-cells of X that are not cells in A. There is a relative boundary homomorphism CW dn : CnCW (X, A) −→ Cn−1 (X, A)

and a short exact sequence of cellular chain complexes i#

j#

0 → C∗CW (A) −→ C∗CW (X) −→ C∗CW (X, A) → 0 .

CHAPTER 1. SINGULAR HOMOLOGY AND COHOMOLOGY

23

((Exercise: Give an explicit description of the relative boundary homomorphism in terms of the maps i∗ , j∗ and/or ∂ of various pairs.)) Introducing coefficients, let M CnCW (X, A; G) = CnCW (X, A) ⊗ G ∼ G = α

and

n CCW (X, A; G) = Hom(CnCW (X, A), G) ∼ =

Y

G,

α

where α runs over the set of n-cells in X that are not cells in A. Using the boundary homomorphisms ∂ = ∂ ⊗ 1 and δ = Hom(∂, 1) we get the cellular homology and cohomology groups HnCW (X, A; G) = Hn (C∗CW (X, A; G), ∂) and n ∗ HCW (X, A; G) = H n (CCW (X, A; G), δ) .

Lemma 1.11.1. There are isomorphisms CnCW (X, A; G) ∼ = Hn (X (n) , X (n−1) ∪ A(n) ; G) and

n CCW (X, A; G) ∼ = H n (X (n) , X (n−1) ∪ A(n) ; G) ,

compatible with the boundary homomorphisms. Theorem 1.11.2 (Cellular (co-)homology). For all CW pairs (X, A) there are isomorphisms Hn (X, A; G) ∼ = HnCW (X, A; G) and

n H n (X, A; G) ∼ (X, A; G) , = HCW

natural with respect to cellular maps of pairs. If X is a CW complex of finite type, meaning that it has only a finite number of n-cells for each n (but may be infinite-dimensional), then the cellular complex C∗CW (X) is finitely generated in each degree. It follows that the cellular homology groups H∗CW (X) are finitely generated in each degree, hence this is also the case for the (isomorphic) singular homology groups H∗ (X). Similarly for CW pairs of (relatively) finite type. Exercise 1.11.3. Prove the lemma and the theorem. Exercise 1.11.4. Consider the CW structure on the unit sphere S k ⊂ Rk+1 , with n-skeleton S n for n ≤ k, and two n-cells en+ and en− for each 0 ≤ n ≤ k. Determine the cellular complex C∗CW (S k ), and compute the cellular homology ∗ groups H∗CW (S k ; G) and the cellular cohomology groups HCW (S k ; G). The cases k = −1 and k = 0 may be treated separately. Be careful with orientations and signs.

CHAPTER 1. SINGULAR HOMOLOGY AND COHOMOLOGY

24

Exercise 1.11.5. Consider the CW structure on the real projective space RP k = S k /∼, where p ∼ −p for p ∈ S k , with n-skeleton RP n for n ≤ k, and one n-cell en for each 0 ≤ n ≤ k. Use naturality for the cellular map f : S k → RP k to determine the cellular complex C∗CW (RP k ), and compute the cellular homology groups H∗CW (RP k ; G) and the cellular cohomology groups ∗ HCW (RP k ; G). You may concentrate on the cases k ≥ 1, G = Z, G = Z/2 and G = Z/p with p an odd prime. Topologists often write Z/m where algebraists might write Z/mZ or Z/(m). Hatcher [2] writes Zm , as topologists used to do, but this is easily confused with the ring of p-adic integers, especially when m = p.

Chapter 2

The universal coefficient theorems There is a natural homomorphism Hn (X) ⊗ G −→ Hn (X; G) taking a tensor product [x] ⊗ g, where x is a singular n-cycle in X, to the homology class [x ⊗ g] of the n-cycle x ⊗ g in the complex C∗ (X; G). This homomorphism is injective, but is not an isomorphism in general. Similarly, there is a natural homomorphism H n (X; G) −→ Hom(Hn (X), G) taking the cohomology class [ϕ] of an n-cocycle ϕ : Cn (X) → G to the homomorphism ϕ∗ : Hn (X) → G, taking [x] to ϕ(x), where x is an n-cycle in X. Note that if x is changed by a boundary ∂y, then ϕ(x) changes by ϕ(∂y) = (δϕ)(y) = 0, since ϕ was assumed to be a cocycle. This homomorphism is surjective, but is not an isomorphism in general. For each pair of spaces (X, A), the tensor product of the long exact sequence in homology with the abelian group G is a chain complex · · · → Hn (A) ⊗ G −→ Hn (X) ⊗ G −→ Hn (X, A) ⊗ G → . . . . but in general this is not an exact complex. On the other hand, the chain complex · · · → Hn (A; G) −→ Hn (X; G) −→ Hn (X, A; G) → . . . . is exact. In this sense, the functor Hn (X; G) is better behaved than the functor Hn (X) ⊗ G. Similarly, applying Hom(−, G) to the long exact sequence in homology we get a cochain complex · · · → Hom(Hn (X, A), G) −→ Hom(Hn (X), G) −→ Hom(Hn (A), G) → . . . . but in general this is not an exact complex. On the other hand, the cochain complex · · · → H n (X, A; G) −→ H n (X; G) −→ H n (A, G) → . . . . 25

CHAPTER 2. THE UNIVERSAL COEFFICIENT THEOREMS

26

is exact. In this sense, the functor H n (X; G) is better behaved than the functor Hom(Hn (X), G).

2.1

Half-exactness

Let

j

i

0 → A −→ B −→ C → 0 be a short exact sequence of abelian groups. Lemma 2.1.1. Let G be an abelian group. Then j⊗1

i⊗1

A ⊗ G −→ B ⊗ G −→ C ⊗ G → 0 is exact, but i ⊗ 1 might not be injective, and 0 → Hom(C, G)

Hom(j,1)

−→

Hom(B, G)

Hom(i,1)

−→

Hom(A, G)

is exact, but Hom(i, 1) might not be surjective. We say that (−) ⊗ G is right exact and that Hom(−, G) is left exact. Proof. ((ETC)) Lemma 2.1.2. The following are equivalent: (a) There is a homomorphism r : B → A with ri = 1 : A → A. (b) There is a homomorphism s : C → B with js = 1 : C → C. Proof. Given r we may choose s so that ir+sj = 1 : B → B, and conversely. In this case, we say that 0 → A → B → C → 0 is a split (short) exact sequence. We call r a retraction and s a section. There are then preferred isomorphisms ∼ = i + s : A ⊕ C −→ B and

∼ =

(j, r) : B −→ A × C . Lemma 2.1.3. If 0 → A → B → C → 0 is split, then i ⊗ 1 : A ⊗ G → B ⊗ G is split injective and Hom(i, 1) : Hom(B, G) → Hom(A, G) is split surjective. Proof. If r : B → A is a retraction, then (r ⊗ 1)(i ⊗ 1) = 1 shows that i ⊗ 1 is (split) injective and Hom(i, 1) Hom(r, 1) = 1 shows that Hom(i, 1) is (split) surjective. Lemma 2.1.4. If C = Z{T } is a free abelian group, then any surjection j : B → C admits a section. Proof. For each basis element t ∈ T , use surjectivity to choose an element s(t) ∈ B with js(t) = t. Since C is free, we can extend additively to obtain the desired homomorphism s : C → B with js = 1. ((Flat, projective.))

CHAPTER 2. THE UNIVERSAL COEFFICIENT THEOREMS

2.2

27

Free resolutions

Taken together, these lemmas show that when applied to short exact sequences 0 → A → B → C → 0 with C a free abelian group, the functors (−) ⊗ G and Hom(−, G) are exact. To control the failure of exactness for general abelian groups C, we resolve C by a free abelian groups. This means that we replace C by a chain complex f2

f1

· · · → F2 −→ F1 −→ F0 → 0 ∼ C and Hn (F∗ , f ) = of free abelian groups Fn = Z{Tn }, such that H0 (F∗ , f ) = 0 for n 6= 0. An isomorphism H0 (F∗ , f ) ∼ = C corresponds to a choice of a homomorphism  : F0 → C that makes the diagram f2

f1



· · · → F2 −→ F1 −→ F0 −→ C → 0 exact at all points. In other words,  is surjective and im(f1 ) = ker(). Such a diagram is called an augmented chain complex. We call the complex (F∗ , f ) a free resolution of C. If we think of C is a chain complex concentrated in degree 0, then  can also be viewed as a chain map  : F∗ → C that induces an isomorphism in homology. A free resolution of C is thus a chain complex of free abelian groups, with homology isomorphic to C (concentrated in degree 0). Any abelian group C admits a short free resolution. For by choosing any set of generators T0 ⊆ C we can let F0 = Z{T0 }, and obtain a surjective augmentation  : F0 → C by sending each element of the basis T0 for F0 to the corresponding element of C. Then ker() ⊆ F0 is a subgroup of a free abelian group. It is an algebraic fact that any subgroup of a free abelian group is again a free group, so there is a set T1 and an isomorphism Z{T1 } ∼ = ker(). We let F1 = Z{T1 } and define f1 : F1 → F0 by sending each generator of T1 to the corresponding element in ker() ⊆ F0 . Then f1



0 → F1 −→ F0 −→ C → 0 (extended by 0’s to the left) is a free resolution of C. Since Fn = 0 for n > 1 we say that this is a resolution of length 1. For example, the free group C = Z has a very short free resolution 

0 → Z −→ Z → 0 with F0 = Z, and a finite cyclic group C = Z/m has a short free resolution m



0 → Z −→ Z −→ Z/m → 0 with F0 = F1 = Z, where f1 multiplies by m. ((Discuss essential uniqueness and functoriality of free resolutions.))

CHAPTER 2. THE UNIVERSAL COEFFICIENT THEOREMS

2.3

28

Tor and Ext

The failure of exactness of the tensor product C⊗G is measured by the homology of the chain complex F∗ ⊗ G obtained by replacing C by a free resolution F∗ . Similarly, the failure of exactness of Hom(C, G) is measured by the cohomology of the cochain complex Hom(F∗ , G). These homology and cohomology groups are called Tor- and Ext-groups. Definition 2.3.1. Let C and G be abelian groups. Choose any free resolution  : F∗ → C of C. The Tor-groups of C and G are the homology groups TorZn (C, G) = Hn (F∗ ⊗ G, f ⊗ 1) for n ≥ 0. The Ext-groups of C and G are the cohomology groups ExtnZ (C, G) = H n (Hom(F ∗ , G), Hom(f, 1)) for n ≥ 0. This is the form of the definition that is interesting in the generality of Rmodules over a general ring R. But in our case, of Z-modules in the guise of abelian groups, only a few of these groups are relevant. Since each abelian group admits a short free resolution 0 → F1 → F0 → C → 0, the Tor-groups are the homology groups of the chain complex f1 ⊗1

0 → F1 ⊗ G −→ F0 ⊗ G → 0 . In view of the right exactness of (−) ⊗ G, we get isomorphisms TorZ0 (C, G) ∼ =C ⊗G and

TorZ1 (C, G) ∼ = ker(f1 ⊗ 1 : F1 ⊗ G → F0 ⊗ G) ,

while TorZn (C, G) = 0 for n ≥ 2. Hence the 0-th Tor-group recovers the tensor product, and the only interesting, new, Tor-group is the 1-st one. We simplify its notation to Tor(C, G) = TorZ1 (C, G) . For example, with C = Z we can use the very short resolution F∗ → Z with F1 = 0, so F∗ ⊗ G is 0 in degree 1 and Tor(Z, G) = 0 for all G. For a more interesting example, with C = Z/m we can use the short free resolution 0 → Z → Z → Z/m → 0, with f1 multiplying with the natural number m, so F∗ ⊗ G is the complex m

0 → G −→ G → 0 concentrated in degrees 1 and 0. The homology in degree 0 is the tensor product G ⊗ Z/m ∼ = G/mG, while the homology in degree 1 is the Tor-group Tor(Z/m, G) = ker(m : G → G) = G[m] . Here we use the notation G[m] = {x ∈ G | mx = 0}

CHAPTER 2. THE UNIVERSAL COEFFICIENT THEOREMS

29

for the exponent m-torsion in G. Another common notation for this subgroup is m G. If G is free abelian, so that (−) ⊗ G is exact, then f1 ⊗ 1 is injective, so that Tor(C, G) = 0 for all C. More generally, we say that G is flat if (−) ⊗ G is exact, and then Tor(C, G) = 0 for all C. An abelian group G is flat if and only if it is torsion free. For instance, G = Q is a torsion free group that is not free. Returning to a general short free resolution 0 → F1 → F0 → C → 0, the Ext-groups are the cohomology groups of the cochain complex 0 → Hom(F0 , G)

Hom(f1 ,1)

−→

Hom(F1 , G) → 0 .

In view of the left exactness of Hom(−, G), we get isomorphisms Ext0Z (C, G) ∼ = Hom(C, G) and

Ext1Z (C, G) ∼ = cok(Hom(f1 , 1) : Hom(F0 , G) → Hom(F1 , G))

(the cokernel of the homomorphism), while ExtnZ (C, G) = 0 for n ≥ 2. Hence the 0-th Ext-group recovers the Hom-group, and the only interesting, new, Extgroup is the 1-st one. We simplify its notation to Ext(C, G) = Ext1Z (C, G) . For example, with C = Z we can use the very short resolution F∗ → Z with F1 = 0, so Hom(F∗ , G) is 0 in degree 1 and Ext(Z, G) = 0 for all G. For a more interesting example, with C = Z/m we can use the short free resolution 0 → Z → Z → Z/m → 0, with f1 multiplying with the natural number m, so Hom(F∗ , G) is the complex m

0 → G −→ G → 0 concentrated in degrees 0 and 1. The cohomology in degree 0 is the Hom-group Hom(Z/m, G) ∼ = G[m], while the cohomology in degree 1 is the Ext-group Ext(Z/m, G) = cok(m : G → G) = G/mG . If G is such that Hom(−, G) is exact, then Hom(f1 , 1) is injective and Ext(C, G) = 0 for all C. In this case we say that G is injective. An abelian group G is injective if and only if it is divisible, meaning that for each x ∈ G and each natural number m there exists an y ∈ G with x = my. For instance, G = Q and G = Q/Z are divisible groups. ((Discuss well-definedness up to coherent preferred isomorphism of Tor- and Ext-groups, and functoriality.))

2.4

The universal coefficient theorem in homology

Theorem 2.4.1 (Universal coefficient theorem). Let (X, A) be a pair of topological spaces, and let G be an abelian group. There is a natural short exact sequence α

0 → Hn (X, A) ⊗ G −→ Hn (X, A; G) → Tor(Hn−1 (X, A), G) → 0 for each n. The sequence is split, but not naturally split.

CHAPTER 2. THE UNIVERSAL COEFFICIENT THEOREMS

30

Proposition 2.4.2. Let (C∗ , ∂) be a chain complex of free abelian groups, and let G be an abelian group. There is a natural short exact sequence α

0 → Hn (C∗ ) ⊗ G −→ Hn (C∗ ⊗ G) → Tor(Hn−1 (C∗ ), G) → 0 for each n. The sequence is split, but not naturally split. Proof. Let Bn = im(∂) ⊆ Zn = ker(∂) ⊆ Cn , as usual. Since each Cn is free, so is each subgroup Bn and Zn . For each n there is a short exact sequence ∂

0 → Zn → Cn −→ Bn−1 → 0 . Since Bn−1 is free, this sequence splits. Tensoring with G, we get a (split) short exact sequence ∂

0 → Zn ⊗ G → Cn ⊗ G −→ Bn−1 ⊗ G → 0 for each n, where we abbreviate ∂ ⊗ 1 to ∂. These fit together in a commutative diagram .. .

.. . 0



/ Cn+1 ⊗ G

/ Zn+1 ⊗ G 0

0

∂



0

∂

∂

/0

 / Bn−1 ⊗ G

/0

0



/ Cn−1 ⊗ G

 .. .

 / Bn ⊗ G 0

 / Cn ⊗ G

/ Zn−1 ⊗ G

0

∂

∂

 / Zn ⊗ G

0

0

∂



0

.. .

∂

 / Bn−2 ⊗ G 0

 .. .

/0

 .. .

which we view as a short exact sequence of chain complexes ∂

0 → (Z∗ ⊗ G, 0) −→ (C∗ ⊗ G, ∂) −→ (B∗−1 ⊗ G, 0) → 0 . Here (Z∗ ⊗ G, 0) denotes the chain complex with Zn ⊗ G in degree n and zero maps as boundary homomorphisms, while (B∗−1 ⊗ G, 0) denotes the chain complex with Bn−1 ⊗ G in degree n and zero boundaries. The associated long exact sequence in homology contains the terms kn−1

k

n Bn ⊗ G −→ Zn ⊗ G −→ Hn (C∗ ⊗ G) −→ Bn−1 ⊗ G −→ Zn−1 ⊗ G .

Hence there is a natural short exact sequence α

0 → cok(kn ) −→ Hn (C∗ ⊗ G) → ker(kn−1 ) → 0 . Chasing the definition of the connecting homomorphism, we see that kn = ιn ⊗1 is equal to the tensor product of the inclusion ιn : Bn ⊆ Zn with G, in each degree n.

CHAPTER 2. THE UNIVERSAL COEFFICIENT THEOREMS

31

By the definition of homology, there is a short exact sequence ι



n n 0 → Bn −→ Zn −→ Hn (C∗ ) → 0 .

Since Bn and Zn are free, this is a short free resolution (F∗ , f ) of the homology group Hn (C∗ ). In the notation used above, F1 = Bn , F0 = Zn and f1 = ιn . Hence the Tor-groups of Hn (C∗ ) and G are the homology groups of the complex F∗ ⊗ G, so that there is an exact sequence ι ⊗1

n 0 → Tor(Hn (C∗ ), G) → Bn ⊗ G −→ Zn ⊗ G → Hn (C∗ ) ⊗ G → 0 .

In other words, there are natural isomorphisms ker(kn ) ∼ = Tor(Hn (C∗ ), G) and cok(kn ) ∼ = Hn (C∗ ) ⊗ G, for all n. Hence we have a natural short exact sequence α

0 → Hn (C∗ ) ⊗ G −→ Hn (C∗ ⊗ G) −→ Tor(Hn−1 (C∗ ), G) → 0 . By inspection of the definitions, the left hand homomorphism α takes [x] ⊗ g in Hn (C∗ ) ⊗ G to [x ⊗ g] in Hn (C∗ ⊗ G), for each n-cycle x ∈ Zn . To see that the universal coefficient short exact sequence admits a splitting, first choose a retraction r : Cn → Zn in the short exact sequence 0 → Zn → Cn → Bn−1 → 0. The composite n ◦ r : Cn → Zn → Hn (C∗ ) defines a chain map (C∗ , ∂) → (H∗ (C∗ ), 0), since r restricts to the identity on Zn , so that n ◦ r is zero on Bn . Tensoring with G we get a chain map (C∗ ⊗ G, ∂) −→ (H∗ (C∗ ) ⊗ G, 0) and an induced map in homology Hn (C∗ ; G) −→ Hn (C∗ ) ⊗ G for each n. This is a retraction for the universal coefficient sequence. Note that we do not claim that the retractions r define a chain map (C∗ , ∂) → (Z∗ , 0).

2.5

The universal coefficient theorem in cohomology

Theorem 2.5.1 (Universal coefficient theorem). Let (X, A) be a pair of topological spaces, and let G be an abelian group. There is a natural short exact sequence β

0 → Ext(Hn−1 (X, A), G) → H n (X, A; G) → Hom(Hn (X, A), G) → 0 for each n. The sequence is split, but not naturally split. Again, this follows from the following result in the case C∗ = C∗ (X, A). Proposition 2.5.2. Let (C∗ , ∂) be a chain complex of free abelian groups, and let G be an abelian group. There is a natural short exact sequence β

0 → Ext(Hn−1 (C∗ ), G) → H n (Hom(C∗ , G)) −→ Hom(Hn (C∗ ), G) → 0 for each n. The sequence is split, but not naturally split.

CHAPTER 2. THE UNIVERSAL COEFFICIENT THEOREMS

32

For an example of the failure of naturality of the splitting, see Section 3.1, Exercise 11 in [2]. Proof. Let Bn = im(∂) ⊆ Zn = ker(∂) ⊆ Cn , as usual. Since each Cn is free, so is each subgroup Bn and Zn . For each n there is a short exact sequence ∂

0 → Zn → Cn −→ Bn−1 → 0 . Since Bn−1 is free, this sequence splits. Forming Hom-groups into G, we get a (split) short exact sequence δ

0 → Hom(Bn−1 , G) −→ Hom(Cn , G) −→ Hom(Zn , G) → 0 for each n, where we abbreviate Hom(∂, 1) to δ. These fit together in a commutative diagram .. .

.. . 0

0

 / Hom(Bn−2 , G)

0

 / Hom(Cn−1 , G)

δ

0

0

 / Hom(Zn−1 , G)

 / Hom(Cn , G)

δ

 / Hom(Zn , G)

 / Hom(Cn+1 , G) δ

 .. .

/0

0

δ

δ

/0

0

δ

0

 / Hom(Bn , G)

0

δ

0

 / Hom(Bn−1 , G)

.. .

 / Hom(Zn+1 , G) 0

 .. .

/0

 .. .

which we view as a short exact sequence of cochain complexes δ

0 → (Hom(B∗−1 , G), 0) −→ (Hom(C∗ , G), δ) −→ (Hom(Z∗ , G), 0) → 0 . Here (Hom(B∗−1 , G), 0) denotes the cochain complex with Hom(Bn−1 , G) in degree n and zero maps as coboundary homomorphisms, while (Hom(Z∗ , G, 0) denotes the cochain complex with Hom(Zn , G) in degree n and zero coboundaries. The associated long exact sequence in cohomology contains the terms kn−1

Hom(Zn−1 , G) −→ Hom(Bn−1 , G) −→ H n (Hom(C∗ , G)) −→ kn

−→ Hom(Zn , G) −→ Hom(Bn , G) . Hence there is a natural short exact sequence β

0 → cok(k n−1 ) → H n (Hom(C∗ , G)) −→ ker(k n ) → 0 . Chasing the definition of the connecting homomorphism, we see that k n = Hom(ιn , 1) is Hom-dual to the inclusion ιn : Bn ⊆ Zn , in each degree n.

CHAPTER 2. THE UNIVERSAL COEFFICIENT THEOREMS

33

By the definition of homology, there is a short exact sequence ι



n n 0 → Bn −→ Zn −→ Hn (C∗ ) → 0 .

Since Bn and Zn are free, this is a short free resolution (F∗ , f ) of the homology group Hn (C∗ ). In the notation used above, F1 = Bn , F0 = Zn and f1 = ιn . Hence the Ext-groups of Hn (C∗ ) and G are the cohomology groups of the cocomplex Hom(F∗ , G), so that there is an exact sequence 0 → Hom(Hn (C∗ ), G) −→ Hom(Zn , G)

Hom(ιn ,1)

−→

Hom(Bn , G) −→ −→ Ext(Hn (C∗ ), G) → 0 .

In other words, there are natural isomorphisms ker(k n ) ∼ = Hom(Hn (C∗ ), G) and cok(k n ) ∼ = Ext(Hn (C∗ ), G), for all n. Hence we have a natural short exact sequence β

0 → Ext(Hn−1 (C∗ ), G) −→ H n (Hom(C∗ , G)) −→ Hom(Hn (C∗ ), G) → 0 . By inspection of the definitions, the right hand homomorphism β takes [ϕ] in H n (Hom(C∗ , G)) to the homomorphism mapping [x] in Hn (C∗ ) to ϕ(x) in G. To see that the universal coefficient short exact sequence admits a splitting, first choose a retraction r : Cn → Zn in the short exact sequence 0 → Zn → Cn → Bn−1 → 0. The composite n ◦ r : Cn → Zn → Hn (C∗ ) defines a chain map (C∗ , ∂) → (H∗ (C∗ ), 0), since r restricts to the identity on Zn , so that n ◦ r is zero on Bn . Hom’ing into G we get a cochain map (Hom(H∗ (C∗ ), G), 0) −→ (Hom(C∗ , G), δ) and an induced map in cohomology Hom(Hn (C∗ ), G) −→ H n (Hom(C∗ , G)) for each n. This is a section for the universal coefficient sequence.

2.6

Some calculations

Consider X = RP 3 with the minimal CW structure, having one cell en in each dimension for 0 ≤ n ≤ 3. The cellular complex C∗ = C∗CW (RP 3 ) is 0

2

0

0 → Z{e3 } −→ Z{e2 } −→ Z{e1 } −→ Z{e0 } → 0 . Hence the integral homology groups are   Z for n = 0      Z/2 for n = 1 Hn (RP 3 ) = 0 for n = 2    Z for n = 3    0 for n > 3.

CHAPTER 2. THE UNIVERSAL COEFFICIENT THEOREMS

34

It follows that   for n = 0, n = 3 G 3 Hn (RP ) ⊗ G = G/2G for n = 1   0 for n = 2, n > 3 and

( G[2] for n = 1 Tor(Hn (RP ) ⊗ G = 0 otherwise. 3

Hence

  G for      G/2G for Hn (RP 3 ; G) ∼ for = G[2]   G for    0 for

n=0 n=1 n=2 n=3 n > 3.

Similarly,   for n = 0, n = 3 G 3 Hom(Hn (RP ), G) = G[2] for n = 1, n > 3   0 for n = 2 and

( G/2G for n = 1 Ext(Hn (RP ), G) = 0 otherwise. 3

Hence

  G for      G[2] for  H n (RP 3 ; G) ∼ = G/2G for    G for    0 for

n=0 n=1 n=2 n=3 n > 3.

For instance,   Z for n = 0      0 for n = 1  n 3 ∼ H (RP ; Z) = Z/2 for n = 2   Z for n = 3    0 for n > 3. Notice how the torsion in Hn (X) is shifted up to H n+1 (X; Z), while the free part of Hn (X) reappears in H n (X; Z). Since H0 (X) is always free, Tor(H0 (X), G) = 0 and Ext(H0 (X), G) = 0, so for n ≤ 1 there are isomorphisms ∼ =

α : Hn (X) ⊗ G −→ Hn (X; G) and

∼ =

β : H n (X; G) −→ Hom(Hn (X), G) . Similarly for relative (co-)homology.

CHAPTER 2. THE UNIVERSAL COEFFICIENT THEOREMS

35

The group Q is torsion free and divisible, so Tor(Hn−1 (X), Q) = 0 and Ext(Hn−1 (X), Q) = 0, and there are isomorphisms ∼ =

α : Hn (X) ⊗ Q −→ Hn (X; Q) and

∼ =

β : H n (X; Q) −→ Hom(Hn (X), Q) for all n. It follows that there is an isomorphism H n (X; Q) ∼ = HomQ (Hn (X; Q), Q) identifying H n (X; Q) with the dual Q-vector space of Hn (X; Q). If H∗ (X) is of finite type, meaning that Hn (X) is finitely generated for each n, then we can write Hn (X) = Tn ⊕ Fn where Tn is a finite abelian group, and Fn is a finitely generated free abelian group. If Fn ∼ = Zr we say that Hn (X) has rank r. Note that Tn ⊗ Q = 0 and Zr ⊗ Q = Qr , so the rank of Hn (X) equals the dimension of Hn (X; Q) ∼ = Hn (X) ⊗ Q as a Q-vector space. The rank of Hn (X) is also known as the n-th Betti number of X.

2.7

Field coefficients

((Recall reduced homology.)) ˜ ∗ (X) = 0 if and only if Proposition 2.7.1. Let X be any space. Then H ˜ ∗ (X; Q) = 0 and H ˜ ∗ (X; Z/p) = 0 for all primes p. H Corollary 2.7.2. A map f : X → Y induces isomorphisms ∼ =

f∗ : H∗ (X) −→ H∗ (Y ) in integral homology if and only if it induces isomorphisms ∼ =

f∗ : H∗ (X; F ) −→ H∗ (Y ; F ) with coefficients in the fields F = Q and F = Z/p, for all primes p. This follows by passage to the mapping cone Cf , using the long exact sequence f∗ ˜ n (Cf ; G) → . . . · · · → Hn (X; G) −→ Hn (Y ; G) −→ H for G = Z and G = F . Proof. The forward implication is clear from the universal coefficient theorem ˜ ∗ (X; Q) = 0 and H ˜ ∗ (X; Z/p) for in homology. For the converse, assume that H all primes p. From the short exact sequence ˜ n (X)/p → H ˜ n (X; Z/p) → H ˜ n−1 (X)[p] → 0 0→H ˜ n (X)/p = 0 and H ˜ n−1 (X)/[p] = 0, so multiplication by p on we deduce that H ˜ ˜ n (X) → H ˜ n (X) ⊗ Q, which inverts every Hn (X) is an isomorphism. Hence H prime, is already an isomorphism. But ˜ n (X) ⊗ Q ∼ ˜ n (X; Q) H =H ˜ n (X) = 0. is zero by assumption, so H

CHAPTER 2. THE UNIVERSAL COEFFICIENT THEOREMS

36

Proposition 2.7.3. Let (X, A) be a pair of topological spaces, and let F be a field. There is a natural isomorphism ∼ =

β : H n (X, A; F ) −→ HomF (Hn (X, A; F ), F ) = Hn (X, A; F )∗ There is a more general version of the universal coefficient theorems, for a principal ideal domain R and an R-module M . Replacing C∗ (X) by C∗ (X; R) one is led to work with a chain complex C∗ of free R-modules. The assumption that R is a PID ensures that the submodules B∗ and Z∗ are still free. This leads to the split short exact sequences 0 → Hn (X; R) ⊗R M → Hn (X; M ) → TorR 1 (Hn−1 (X; R), M ) → 0 and 0 → Ext1R (Hn−1 (X; R), M ) → H n (X; M ) → HomR (Hn (X; R), M ) → 0 , and similarly for relative (co-)homology. In the case where R is a field F the 1 derived functors TorF 1 and ExtF vanish. This leads to the stated isomorphism. ˜ ∗ (X) = 0 if and only if H ˜ ∗ (X; Q) = 0 and H ˜ ∗ (X; Z/p) = 0 Corollary 2.7.4. H for all primes p.

Chapter 3

Cup product We turn to a method of introducing a product structure on C ∗ (X; R) and H ∗ (X; R), for R a ring, induced from the diagonal map ∆ : X → X × X.

3.1

The Alexander–Whitney diagonal approximation

Let k, ` ≥ 0. Inside the standard (k + `)-simplex ∆k+` = [v0 , . . . , vk , . . . , vk+` ] there is a front k-simplex ∆k ∼ = [v0 , . . . , vk ] ⊂ ∆k+` where tk+1 = · · · = tk+` = 0, and a back `-simplex ∆` ∼ = [vk , . . . , vk+` ] ⊂ ∆k+` where t0 = · · · = tk−1 = 0. These meet in the single vertex vk , where tk = 1. Let λkk+` : ∆k → ∆k+` and ρ`k+` : ∆` → ∆k+` be the two affine linear embeddings. To each singular (k + `)-simplex σ : ∆k+` → X in a topological space X, we can associate the front k-face σλkk+` = σ|[v0 , . . . , vk ] : ∆k → X and the back `-face σρ`k+` = σ|[vk , . . . , vk+` ] . : ∆` → X Their tensor product defines a homomorphism Ψk,` : Ck+` (X) −→ Ck (X) ⊗ C` (X) that takes σ to σλkk+` ⊗ σρ`k+` = σ|[v0 , . . . , vk ] ⊗ σ|[vk , . . . , vk+` ] . 37

CHAPTER 3. CUP PRODUCT

38

For k = ` = 0, the homomorphism Ψ0,0 : C0 (X) → C0 (X) ⊗ C0 (X) corresponds to the diagonal map ∆ : X → X × X taking a point p ∈ X to (p, p) ∈ X × X, under the correspondences C0 (X) ∼ = = Z{X} and C0 (X) ⊗ C0 (X) ∼ Z{X} ⊗ Z{X} ∼ = Z{X × X}. For two chain complexes (C∗ , ∂) and (D∗ , ∂), we define the tensor product chain complex (C∗ ⊗ D∗ , ∂) to be given in degree n by M (C × D)n = Ck ⊗ D` k+`=n

with boundary homomorphism given by ∂(x ⊗ y) = ∂x ⊗ y + (−1)k x ⊗ ∂y for x ∈ Ck and y ∈ D` . Note that ∂ 2 (x ⊗ y) = ∂(∂x ⊗ y + (−1)k x ⊗ ∂y) = ∂ 2 x ⊗ y + (−1)k−1 ∂x ⊗ ∂y + (−1)k ∂x ⊗ ∂y + (−1)2k−1 x ⊗ ∂ 2 y =0 so that (C∗ ⊗D∗ , ∂) is a chain complex. The sign (−1)k can be justified geometrically, since we are commuting the passage to a boundary past the k-dimensional object k, or algebraically, to make sure that the two middle terms in the above sum cancel. For each n ≥ 0, we can form the sum over all (k, `) with k + ` = n of the homomorphisms Ψk,` , to get the homomorphism M Ψn : Cn (X) −→ Ck (X) ⊗ C` (X) k+`=n

taking σ : ∆n → X to M

= σ|[v0 , . . . , vk ] ⊗ σ|[vk , . . . , vk+` ] .

k+`=n

Lemma 3.1.1. The identity Ψk,` ◦ ∂ = (∂ ⊗ 1)Ψk+1,` + (−1)k (1 ⊗ ∂)Ψk,`+1 holds, so the homomorphisms (Ψn )n define a chain map Ψ# : C∗ (X) −→ C∗ (X) ⊗ C∗ (X) . Since Ψ0 is compatible with the diagonal map, we call Ψ# a diagonal approximation. Proof. We must prove that the diagram Cn+1 (X)

∂

Ψn+1

 (C∗ (X) ⊗ C∗ (X))n+1

/ Cn (X) Ψn

∂

 / (C∗ (X) ⊗ C∗ (X))n

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39

commutes, for each n. We check that for each σ : ∆n+1 → X and each pair (k, `) with k + ` = n, the images of σ under Ψn ∂ and ∂Ψn+1 have the same components in Ck (X) ⊗ C` (X). The (k, `)-th component of ∂Ψn (σ) is the sum of two contributions. One comes from the composite (∂ ⊗ 1)Ψk+1,` : Cn+1 (X) → Ck+1 (X) ⊗ C` (X) → Ck (X) ⊗ C` (X) and the other comes from the composite (−1)k (1 ⊗ ∂)Ψk,`+1 : Cn+1 (X) → Ck (X) ⊗ C`+1 (X) → Ck (X) ⊗ C` (X) . The first takes σ : ∆n+1 → X to (∂ ⊗ 1)σ|[v0 , . . . , vk+1 ] ⊗ σ|[vk+1 , . . . , vn+1 ] =

k+1 X

(−1)i σ|[v0 , . . . , vˆi , . . . , vk+1 ] ⊗ σ|[vk+1 , . . . , vn+1 ]

i=0

and the second takes σ to (−1)k (1 ⊗ ∂)σ|[v0 , . . . , vk ] ⊗ σ|[vk , . . . , vn+1 ] = (−1)k

`+1 X

(−1)j σ|[v0 , . . . , vk ] ⊗ σ|[vk , . . . , vˆk+j , . . . , vn+1 ]

j=0

=

n+1 X

(−1)i σ|[v0 , . . . , vk ] ⊗ σ|[vk , . . . , vˆi , . . . , vn+1 ]

i=k

Notice that the term i = k + 1 in the first sum is equal to the term i = k in the second sum, up to a sign. Hence these two terms cancel when we add the expressions together, so that the (k, `)-th component of ∂Ψn (σ) is k X

(−1)i σ|[v0 , . . . , vˆi , . . . , vk+1 ] ⊗ σ|[vk+1 , . . . , vn+1 ]

i=0

+

n+1 X

(−1)i σ|[v0 , . . . , vk ] ⊗ σ|[vk , . . . , vˆi , . . . , vn+1 ] .

i=k+1

On the other hand, the (k, `)-th component of Ψn ∂(σ) is n+1 X

Ψk,` (

(−1)i σ|[v0 , . . . , vˆi , . . . , vn+1 ])

i=0

=

k X

(−1)i σ|[v0 , . . . , vˆi , . . . , vk+1 ] ⊗ σ|[vk+1 , . . . , vn+1 ]

i=0

+

n+1 X

(−1)i σ|[v0 , . . . , vk ] ⊗ σ|[vk , . . . , vˆi , . . . , vn+1 ] .

i=k+1

These expressions are the same, proving the claim.

CHAPTER 3. CUP PRODUCT

3.2

40

The cochain cup product

Let R be a ring, and consider cochains and cohomology with coefficients in R. The cochain cup product is a pairing ∪

C k (X; R) ⊗R C ` (X; R) −→ C k+` (X; R) . For cochains ϕ : Ck (X) → R and ψ : C` (X) → R the cup product is defined to be the (k + `)-cochain ϕ ∪ ψ : Ck+` (X) → R given as the composite Ψk,`

ϕ⊗ψ

·

Ck+` (X) −→ Ck (X) ⊗ C` (X) −→ R ⊗ R −→ R where Ψk,` is as in the previous subsection and · : R ⊗ R → R is the ring multiplication. More explicitly, the cup product ϕ ∪ ψ takes the value (ϕ ∪ ψ)(σ) = ϕ(σ|[v0 , . . . , vk ]) · ϕ(σ|[vk , . . . , vk+` ]) on a (k + `)-simplex σ : ∆k+` → X. Lemma 3.2.1. The cochain cup product is unital and associative, with unit element 1 ∈ C 0 (X; R) the cochain  : C0 (X) → R that sends each 0-simplex to the ring unit 1 ∈ R. A graded ring is a graded abelian group A∗ = (An )n with a unital and associative pairing Ak ⊗ A` → Ak+` for all k, `, which we can also write as a homomorphism A∗ ⊗ A∗ → A∗ . By the lemma above the cochains C ∗ (X; R) constitute a graded ring.

3.3

The cohomology cup product

The cochain cup product satisfies a Leibniz formula. Lemma 3.3.1. The identity δ(ϕ ∪ ψ) = δϕ ∪ ψ + (−1)k ϕ ∪ δψ holds in C k+`+1 (X; R), for ϕ ∈ C k (X; R) and C ` (X; R), so the cup product defines a cochain map ∪ : C ∗ (X; R) ⊗R C ∗ (X; R) −→ C ∗ (X; R) . Proof. Let n = k + `. For each (n + 1)-simplex σ, we have δ(ϕ ∪ ψ)(σ) = (ϕ ∪ ψ)(∂σ) = (ϕ ⊗ ψ)(Ψk,` ◦ ∂)(σ)

CHAPTER 3. CUP PRODUCT

41

which by the lemma of the previous subsection is the sum of (ϕ ⊗ ψ)(∂ ⊗ 1)Ψk+1,` (σ) = (δϕ ⊗ ψ)Ψk+1,` (σ) = (δϕ ∪ ψ)(σ) and (−1)k (ϕ ⊗ ψ)(1 ⊗ ∂)Ψk,`+1 (σ) = (−1)k (ϕ ⊗ δψ)Ψk,`+1 (σ) = (−1)k (ϕ ∪ δψ)(σ) .

Corollary 3.3.2. If ϕ ∈ C k (X; R) and ψ ∈ C ` (X; R) are cocycles, then ϕ∪ψ ∈ C k+` (X; R) is a cocycle. If furthermore ϕ is a coboundary, or ψ is a coboundary, then ϕ ∪ ψ is a coboundary. Proof. If δϕ = 0 and δψ = 0 then δ(ϕ ∪ ψ) = 0 ∪ ψ + (−1)k ϕ ∪ 0 = 0. If also ϕ = δξ then δ(ξ ∪ ψ) = ϕ ∪ ψ + ξ ∪ 0 = ϕ ∪ ψ. If instead ψ = δη then δ(ϕ ∪ η) = 0 ∪ η + (−1)k ϕ ∪ ψ = (−1)k ϕ ∪ ψ. The cohomology cup product is the induced pairing ∪

H k (X; R) ⊗R H ` (X; R) −→ H k+` (X; R) given by the formula [ϕ] ∪ [ψ] = [ϕ ∪ ψ] for each k-cocycle ϕ and each `-cocycle ψ. It is well-defined by the corollary above. The cup product makes H ∗ (X; R) a graded ring. Lemma 3.3.3. The cohomology cup product is unital and associative, with unit element 1 ∈ H 0 (X; R) the cohomology class of the cocycle  : C0 (X) → R that sends each 0-simplex to the ring unit 1 ∈ R. A cup product for simplicial cohomology can be defined by the same formula as for singular cohomology. Hence the isomorphism between singular cohomology and simplicial cohomology is compatible with the cup products, so that for simplicial complexes, or more generally, for ∆-complexes, the cup products in singular cohomology can be computed using simplicial cochains. Example 3.3.4. The closed orientable surface Mg of genus g ≥ 1 has a triangulation as a ∆-complex obtained by triangulating a regular 4g-gon by starring with an interior point, and identifying the boundary edges pairwise according to the pattern −1 −1 a1 , b1 , a1−1 , b−1 1 , . . . , ag , bg , ag , bg The integral homology groups are H0 (Mg ) = Z, H1 (Mg ) = Z{a1 , b1 , . . . , ag , bg } and H2 (Mg ) ∼ = Z. A generator of H2 (Mg ) is represented by the 2-cycle given by the signed sum of all of the 2-simplices in the triangulation, with sign +1 for the 2-simplices spanned by the center and one of the positively oriented edges ai or bj , and sign −1 for the 2-simplices spanned by the center and one of the negatively oriented edges a−1 or b−1 i j .

CHAPTER 3. CUP PRODUCT

42

Dually, the integral cohomology groups are H 0 (Mg ) = H 0 (Mg ; Z) = Z{1}, H 1 (Mg ) = H 1 (Mg ; Z) = Z{α1 , β1 , . . . , αg , βg } and H 2 (Mg ) = H 2 (Mg ; Z) ∼ = Z{γ}, with αi dual to ai and βi dual to bi . ((Recall what duality means for a basis.)) A generator γ ∈ H 2 (Mg ) is represented by a 2cochain/cocycle that takes the value +1 on the 2-cycle representing a generator of H2 (Mg ). The interesting cup product is the pairing ∪ : H 1 (Mg ) ⊗ H 1 (Mg ) −→ H 2 (Mg ) . To compute cup products, we must represent the cohomology classes αi and βj by 1-cocycles, say ϕi and ψj . The condition δϕi = 0 asserts that the alternating sum of values of ϕi on the three edges of each 2-simplex in Mg must be 0. To represent αi , ϕi must evaluate to 1 on the edge ai . By inspection, we can let ϕi evaluate to 1 on the two edges leading from the center to the end-points of ai and a−1 i , and to 0 on all other edges. Similarly, ψj evaluates to 1 on the edge bj , as well as on the two edges leading from the center to the end-points of bj and b−1 j , and to 0 on all other edges. The cup product of two 1-cocycles ϕ and ψ is the 2-cocycle whose value on a 2-simplex σ is the product (ϕ ∪ ψ)(σ) = ϕ(σ|[v0 , v1 ]) · ψ(σ|[v1 , v2 ]) . The 2-simplices of Mg fall into g groups of four triangles each. The cocycles ϕi and ψi are zero outside of the i-th group, so if i 6= j the cup product of ϕi or ψi with ϕj or ψj is zero on all 2-simplices. Hence these cup products are zero on the simplicial cochain level, and ai ∪ aj = ai ∪ bj = bi ∪ aj = bi ∪ bj = 0 for i 6= j. Fortunately, the case i = j is more interesting. The cup product ϕi ∪ ψi takes the value ϕi (ai ) · ψi (bi ) = 1 · 1 = 1 on the 2-simplex spanned by the center and the edge bi , and is zero on the other 2-simplices. Hence this cup product evaluates to +1 on the 2-cycle representing the generator of H2 (Mg ), so the cohomology cup product αi ∪ βi = γ equals the dual generator of H 2 (Mg ). The cup product ψi ∪ ϕi takes the value ψi (bi ) · ϕi (ai ) = 1 · 1 = 1 on the 2-simplex spanned by the center and the edge a−1 i , and is zero on the other 2-simplices. Hence this cup product evaluates to −1 on the 2-cycle representing the generator of H2 (Mg ), so the cohomology cup product βi ∪ αi = −γ

CHAPTER 3. CUP PRODUCT

43

equals the negative of the dual generator of H 2 (Mg ). The cup products ϕi ∪ ϕi and ψi ∪ ψi are zero on all 2-simplices, so the cohomology cup products αi ∪ αi and βi ∪ βi are both zero. The bilinear pairing H 1 (Mg ) × H 1 (Mg ) → H 2 (Mg ) is thus identified with the bilinear pairing Z2g × Z2g → Z represented by the skew-symmetric 2g × 2g matrix   0 1 ... 0 0 −1 0 . . . 0 0    .. .. . . .. ..   . . . . .    0 0 ... 0 1 0 0 . . . −1 0   0 1 with g copies of the hyperbolic form along the diagonal, and zeroes −1 0 elsewhere. With a different choice of basis, the cup product pairing corresponds to a different matrix. The natural choice made corresponds to a matrix that has as many vanishing entries as is possible. We assumed that g ≥ 1. The conclusion holds as stated in the case g = 0 with M0 = S 2 , in a somewhat trivial way.

3.4

Relative cup products, naturality

Let A, B ⊆ X be subspaces. If ϕ ∈ C k (X, A; R) vanishes on chains in A and ψ ∈ C ` (X, B; R) vanishes on chains in B, then ϕ ∪ ψ ∈ C k+` (X, A + B; R) vanishes on chains in A or in B. Hence there is a relative cup product ∪

H k (X, A; R) ⊗R H ` (X, B; R) −→ H k+` (X, A + B; R) . If {A, B} is excisive, so that H∗ (A + B) → H∗ (A ∪ B) is an isomorphism, then H ∗ (X, A ∪ B; R) → H ∗ (X, A + B; R) is an isomorphism by the long exact sequence and universal coefficient theorem. (This applies, for instance, when A and B are open subsets, or X is a CW complex and A and B are subcomplexes.) Then the cup product lifts through the isomorphism to ∪

H k (X, A; R) ⊗R H ` (X, B; R) −→ H k+` (X, A ∪ B; R) . Some important special cases are the relative cup products ∪

H k (X, A; R) ⊗R H ` (X; R) −→ H k+` (X, A; R) ∪

H k (X; R) ⊗R H ` (X, A; R) −→ H k+` (X, A; R) ∪

H k (X, A; R) ⊗R H ` (X, A; R) −→ H k+` (X, A; R) for all pairs (X, A). For each map f : X → Y the cup product satisfies f # (ϕ ∪ ψ) = f # (ϕ) ∪ f # (ψ)

CHAPTER 3. CUP PRODUCT

44

in C k+` (X; R), for ϕ ∈ C k (Y ; R) and ψ ∈ C ` (Y ; R), so the cochain cup product is natural in the sense that the diagram C k (Y ; R) ⊗R C ` (Y ; R)

∪

/ C k+` (Y ; R)

∪

 / C k+` (X; R)

f # ⊗f #

 C k (X; R) ⊗R C ` (X; R)

f#

commutes. It follows that the cohomology cup product satisfies f ∗ (α ∪ β) = f ∗ (α) ∪ f ∗ (β) in H k+` (X; R), for α ∈ H k (X; R) and β ∈ H ` (X; R), so the cohomology cup product is natural in the same way. Hence the graded ring H ∗ (X; R) is functorial for all spaces X. Similarly, the relative cohomology ring H ∗ (X, A; R) is functorial for all pairs (X, A).

3.5

Cross product

For a pair of spaces X and Y , the natural projection maps p1 : X × Y −→ X p2 : X × Y −→ Y induce cochain homomorphisms ∗ ∗ p# 1 : C (X; R) −→ C (X × Y ; R) ∗ ∗ p# 2 : C (Y ; R) −→ C (X × Y ; R) .

When combined with the cochain cup product for X × Y we get a pairing p# ⊗p#

× : C k (X; R) ⊗R C ` (Y ; R) 1−→2 C k (X × Y ; R) ⊗R C ` (X × Y ; R) ∪

−→ C k+` (X × Y ; R) called the cochain cross product, denoted ×. It takes a k-cocycle ϕ : Ck (X) → R on X and an `-cocycle ψ : C` (Y ) → R on Y to the cup product of their respective pullbacks to X × Y : # ϕ × ψ = p# 1 (ϕ) ∪ p2 (ψ) . Its value on a (k + `)-simplex (σ, τ ) : ∆k+` → X × Y is, by definition, (ϕ × ψ)(σ, τ ) = ϕ(σ|[v0 , . . . , vk ]) · ψ(τ |[vk , . . . , vk+` ]) . Lemma 3.5.1. The identity δ(ϕ × ψ) = δϕ × ψ + (−1)k ϕ × δψ holds in C k+`+1 (X×Y ; R), for ϕ ∈ C k (X; R) and C ` (Y ; R), so the cross product defines a cochain map × : C ∗ (X; R) ⊗R C ∗ (Y ; R) −→ C ∗ (X × Y ; R) .

CHAPTER 3. CUP PRODUCT

45

Proof. This follows from the cup product Leibniz formula by naturality: # δ(ϕ × ψ) = δ(p# 1 (ϕ) ∪ p2 (ψ)) # # k # = δp# 1 (ϕ) ∪ p2 (ψ) + (−1) p1 (ϕ) ∪ δp2 (ψ) # # k # = p# 1 (δϕ) ∪ p2 (ψ) + (−1) p1 (ϕ) ∪ p2 (δψ)

= δϕ × ψ + (−1)k ϕ ∪ δψ

The cohomology cross product is the induced pairing ×

H k (X; R) ⊗R H ` (Y ; R) −→ H k+` (X × Y ; R) given by the formula [ϕ] × [ψ] = [ϕ × ψ] for each k-cocycle ϕ in X and each `-cocycle ψ in Y . It is well-defined by the cross product Leibniz formula. The cross product can be computed in terms of the cup product by the formula α × β = p∗1 (α) ∪ p∗2 (β) . To compute the cross product in some interesting examples, we must first discuss some of its formal properties.

3.6

Relative cross products, naturality

Let (X, A) and (Y, B) be pairs. If ϕ ∈ C k (X, A; R) vanishes on chains in A and ψ ∈ C ` (Y, B; R) vanishes on chains in B, then ϕ × ψ ∈ C k+` (X × Y, A × Y + X × B; R) vanishes on chains in A × Y or in X × B. Hence there is a relative cross product ×

H k (X, A; R) ⊗R H ` (Y, B; R) −→ H k+` (X × Y, A × Y + X × B; R) . If A and B are open, or if X and Y are CW complexes and A and B are subcomplexes, then H∗ (A×Y +X ×B) → H∗ (A×Y ∪X ×B) is an isomorphism, so H ∗ (X × Y, A × Y ∪ X × B; R) ∼ = H ∗ (X × Y, A × Y + X × B; R). Then the cross product lifts to ×

H k (X, A; R) ⊗R H ` (Y, B; R) −→ H k+` (X × Y, A × Y ∪ X × B; R) . The target group is often written as H k+` ((X, A)×(Y, B); R), using the notation (X, A) × (Y, B) = (X × Y, A × Y ∪ X × B) . As regards naturality, for each pair of maps f : X → X 0 and g : Y → Y 0 there is a map f × g : X × Y → X 0 × Y 0 , and the cochain cross product satisfies (f × g)# (ϕ × ψ) = f # (ϕ) × g # (ψ)

CHAPTER 3. CUP PRODUCT

46

in C k+` (X × Y ; R), for ϕ ∈ C k (X 0 ; R) and ψ ∈ C ` (Y 0 ; R). Hence the cohomology cross product satisfies (f × g)∗ (α × β) = f ∗ (α) × g ∗ (β) in H k+` (X × Y ; R), for α ∈ H k (X 0 ; R) and β ∈ H ` (Y 0 ; R), and the diagram ×

H k (X 0 ; R) ⊗R H ` (Y 0 ; R)

/ H k+` (X 0 × Y 0 ; R)

f ∗ ⊗g ∗

 H k (X; R) ⊗R H ` (Y ; R)

×



(f ×g)∗

/ H k+` (X × Y ; R)

commutes. Similarly, the relative cross product is natural for all pairs (X, A) and (Y, B). Note that the cup product can be recovered from the cross product, by pullback along the diagonal map ∆ : X → X × X. The composite ∆∗

×

H k (X; R) ⊗R H ` (X; R) −→ H k+` (X × X; R) −→ H k+` (X; R) is equal to the cup product, since p1 ∆ = 1 = p2 ∆, so ∆∗ p∗1 = 1 = ∆∗ p∗2 and ∆∗ (α × β) = ∆∗ (p∗1 (α) ∪ p∗2 (β)) = ∆∗ p∗1 (α) ∪ ∆∗ p∗2 (β) = α ∪ β . The same result holds at the cochain level. Naturality with respect to the connecting homomorphisms is a bit more subtle. Lemma 3.6.1. For all pairs (X, A) and spaces Y the natural square H k (A; R) ⊗R H ` (Y ; R)

×

/ H k+` (A × Y ; R)

δ⊗1

 H k+1 (X, A; R) ⊗R H ` (Y ; R)

×



δ

/ H k+`+1 (X × Y, A × Y ; R)

commutes, so δ(α × η) = δα × η k

`

for α ∈ H (A; R) and η ∈ H (Y ; R). See [2, p. 210] for the proof. Lemma 3.6.2. For all spaces X and pairs (Y, B) the natural square H k (X; R) ⊗R H ` (B; R)

×

/ H k+` (X × B; R)

1⊗δ

 H k (X; R) ⊗R H `+1 (Y, B; R)

×

δ

/ H k+`+1 (X × Y, X × B; R)

commutes up to the sign (−1)k , so δ(ξ × β) = (−1)k ξ × δβ for ξ ∈ H k (X; R) and β ∈ H ` (B; R).



CHAPTER 3. CUP PRODUCT

47

Proof. Let ϕ ∈ C k (X; R) and ψ ∈ C ` (B; R) be cocycles representing ξ and η, respectively. Choose an extension ψ˜ ∈ C ` (Y ; R) of ψ. Then ϕ × ψ˜ ∈ C k+` (X × Y ; R) is an extension of ϕ × ψ ∈ C k+` (X × B), and δ(ξ × β) is represented by ˜ ∈ C k+`+1 (X × Y, X × B; R). Since ϕ is a cocycle of degree k, this δ(ϕ × ψ) equals (−1)k ϕ × δ ψ˜ by the Leibniz formula, which represents (−1)k ξ ⊗ δβ. Recall that H 1 (I, ∂I; R) = R{α}, where α is dual to the generator of H1 (I, ∂I; R) represented by the 1-cycle ∆1 ∼ = I. The other cohomology groups H m (I, ∂I; R) vanish. Lemma 3.6.3. Let Y be any space. The cross product ×

H 1 (I, ∂I; R) ⊗R H n−1 (Y ; R) −→ H n (I × Y, ∂I × Y ; R) is an isomorphism. Hence each element of H n (I × Y, ∂I × Y ; R) can be written uniquely as α × β, where β ∈ H n−1 (Y ; R). Similarly for pairs (Y, B). Proof. The long exact sequence in cohomology for the pair (I ×Y, ∂I ×Y ) breaks up into short exact sequences, since the inclusion ∂I → I admits a section up to homotopy. Similarly for the pair (I, ∂I). By naturality of the cross product, and flatness of H 1 (I, ∂I; R) = R{α}, we have a map of vertical short exact sequences 0

0  H 0 (I; R) ⊗R H n−1 (Y ; R)

×

 / H n−1 (I × Y ; R)

 H 0 (∂I; R) ⊗R H n−1 (Y ; R)

×

 / H n−1 (∂I × Y ; R)

×

 / H n (I × Y, ∂I × Y ; R)

δ⊗1

 H 1 (I, ∂I; R) ⊗R H n−1 (Y ; R)

δ

 0

 0

It is clear from unitality and a decomposition ∂I × Y ∼ = Y t Y that the upper and middle cross product maps are isomorphisms, hence so is the lower cross product. Let k, ` ≥ 0 and n = k + `. Note that (I k , ∂I k ) × (I ` , ∂I l ) = (I n , ∂I n ), since I × I ` = I n and ∂I k × I ` ∪ I k × ∂I ` = ∂I n . k

Corollary 3.6.4. For k, ` ≥ 0 and n = k + `, the cross product ×

H k (I k , ∂I k ; R) ⊗R H ` (I ` , ∂I l ; R) −→ H n (I n , ∂I n ; R) is an isomorphism. Hence H n (I n , ∂I n ; R) is the free R-module generated by the n-fold cross product α × ··· × α where α ∈ H 1 (I, ∂I; R) is the standard generator. The remaining cohomology groups H m (I n , ∂I n ; R) are zero.

CHAPTER 3. CUP PRODUCT

48

Recall that H 0 (S 1 ; R) = R{1} and H 1 (S 1 ; R) = R{α}, where α is dual to the generator of H1 (S 1 ; R) represented by the 1-cycle ∆1 → ∆1 /∂∆1 ∼ = S1. m 1 The other cohomology groups H (S ; R) vanish. Proposition 3.6.5. Let Y be any space. The cross product ×

H ∗ (S 1 ; R) ⊗R H ∗ (Y ; R) −→ H ∗ (S 1 × Y ; R) is an isomorphism, and similarly for pairs (Y, B). Hence each element of H n (S 1 × Y ; R) can be written uniquely as a sum α × β + 1 × γ, with β ∈ H n−1 (Y ; R) and γ ∈ H n (Y ; R). Proof. We use the pushout square ∂I

/I

 ?

 / S1

where ? = {s0 } is the base-point of S 1 . The map (I, ∂I) → (S 1 , ?) induces a cohomology isomorphism, and similarly when multiplied by Y . In view of the commutative square H 1 (S 1 , ?; R) ⊗R H n−1 (Y ; R)  H 1 (I, ∂I; R) ⊗R H n−1 (Y ; R)

×

/ H n (S 1 × Y, ? × Y ; R)

×

 / H n (I × Y, ∂I × Y ; R)

and the previous lemma, it follows that the upper cross product is an isomorphism. The long exact sequence for the pair (S 1 × Y, ? × Y ) also breaks up, since the inclusion ? → S 1 admits a retraction, and we have another map of vertical short exact sequences 0

0

 H 1 (S 1 , ?; R) ⊗R H n−1 (Y ; R)

×

 [H ∗ (S 1 ; R) ⊗R H ∗ (Y ; R)]n  H 0 (?; R) ⊗R H n (Y ; R)  0

×

×

 / H n (S 1 × Y, ? × Y ; R)  / H n (S 1 × Y ; R)  / H n (? × Y ; R)  0

We have seen that the upper cross product is an isomorphism. Since the lower one is obviously an isomorphism, it follows that the middle map is also an isomorphism.

CHAPTER 3. CUP PRODUCT

49

Example 3.6.6. Let T n = S 1 ×· · ·×S 1 be the n-dimensional torus. The n-fold cross product ×

H ∗ (S 1 ; R) ⊗R · · · ⊗R H ∗ (S 1 ; R) −→ H ∗ (T n ; R) is an isomorphism. Hence H k (T n ; R) is a free R-module with basis the set of k-fold cup products αi1 ∪ · · · ∪ αik 1 n for 1 ≤ i1 < · · · < ik ≤ n, where αi = p# i (α) ∈ H (T ; R) is the pullback of the 1 1 generator α ∈ H (S ; R) along the i-th projection map pi : T n → S 1 . This is clear by induction on n, using the proposition above, which tells us that a basis is given by the set of n-fold cross products

β1 × · · · × βn ∈ H k (T n ; R) where k of the classes βi are equal to α, and the remaining (n − k) of the classes βi are equal to 1. Numbering the βi that are equal to α as βi1 , . . . , βik , we get the asserted formula.

3.7

Projective spaces

S Let RP n be the n-dimensional real projective space, and let RP ∞ = n RP n . Recall that the cellular complex C∗CW (RP n ) has one generator ek in each degree 0 ≤ k ≤ n, with boundary homomorphism ∂(ek ) = (1 + (−1)k )ek−1 . Hence ∗ CCW (RP n ; Z/2) has trivial coboundary, so H k (RP n ; Z/2) ∼ = Z/2 for each 0 ≤ k ≤ n, where the generator in degree k evaluates to 1 ∈ Z/2 on ek . Proposition 3.7.1. H ∗ (RP n ; Z/2) ∼ = Z/2[x]/(xn+1 ) and

H ∗ (RP ∞ ; Z/2) ∼ = Z/2[x] ,

where |x| = 1. Proof. We simplify notation by writing P n for RP n and omitting the coefficient ring Z/2. By induction on n and naturality with respect to the inclusions P n−1 → P n → P ∞ , it suffices to prove that the cup product of a generator of H n−1 (P n ) and a generator of H 1 (P n ) is a generator of H n (P n ), for n ≥ 2. It is no more difficult to prove that the cup product ∪

H i (P n ) ⊗ H j (P n ) −→ H n (P n ) is an isomorphism, for i + j = n. Consider the subspaces Ri+1+0 ⊂ Ri+1+j ⊃ R0+1+j , which meet in R0+1+0 . Passing to the spaces of lines through the origin we have the subspaces P i ⊂ P n ⊃ P j meeting in a single point P i ∩ P j = {q}. Inside P n we have an affine n-space Rn ∼ = U ⊂ P n where the i-th coordinate is nonzero (counting from 0 to i + j = n), whose complement is a copy of P n−1 . The intersection U ∩ P i ∼ = Ri i−1 i is an affine i-space, with complement P in P . Similarly, the intersection U ∩ Pj ∼ = Rj is an affine j-space, with complement P j−1 in P j .

CHAPTER 3. CUP PRODUCT

50

We have a commutative diagram H i (P n ) ⊗ H j (P n ) O

/ H n (P n ) O

∪

H i (P n , P n − P j ) ⊗ H j (P n , P n − P i )

∪

/ H n (P n , P n − {q})

∼ =

∼ =

  ∪ / H n (Rn , Rn − {q}) H i (Rn , Rn − Rj ) ⊗ H j (Rn , Rn − Ri ) O hh3 hhhh h h ∗ ∗ h p1 ⊗p2 hh hhhh × hhhh

H i (Ri , Ri − {0}) ⊗ H j (Rj , Rj − {0})

The downward arrows are isomorphisms by excision. The cross product is an isomorphism by earlier calculations (replacing (Rn , Rn −{0}) by (I n , ∂I n ), etc.). The projection p1 : (Rn , Rn − Rj ) → (Ri , Ri − {0}) away from a copy of Rj is a homotopy equivalence, and similarly for p2 , so p∗1 and p∗2 are isomorphisms. This proves that the middle horizontal cup product is an isomorphism. The rest is maneuvering from the relative to the absolute case. The complement P n − P j deformation retracts to P i−1 , since it consists of points [x0 : · · · : xn ] where at least one of the homogeneous coordinates x0 , . . . , xi−1 is nonzero, and a deformation retraction to the subspace P i−1 , where all of the homogeneous coordinates xi , . . . , xn are zero, is given by the formula (t, [x0 : · · · : xn ]) 7→ [x0 : · · · : xi−1 : txi : · · · : txn ] . Hence the homomorphism H i (P n , P n − P j ) → H i (P n ) factors as the composite H i (P n , P n − P j ) −→ H i (P n , P j−1 ) −→ H i (P n ) where the first arrow is an isomorphism because of the deformation retraction, and the second arrow is an isomorphism by consideration of the cellular complexes. The same conclusion holds for i replaced by j or n. Hence the upper vertical arrows in the big diagram are isomorphisms, so that the upper horizontal cup product is an isomorphism. Let x ∈ H 1 (P n ) be the generator. Let xn = x ∪ · · · ∪ x ∈ H n (P n ) denote the n-th cup power. By induction on n, we know that xn−1 ∈ H n−1 (P n ) restricts to the generator of H n−1 (P n−1 ), hence is the generator of H n−1 (P n ). By what we have just shown, xn = x ∪ xn−1 is the generator of H n (P n ). We return to integer coefficients. Let CP n be the n-dimensional complex S projective space, of real dimension 2n, and let CP ∞ = n CP n . The cellular complex C∗CW (CP n ) has one generator e2k in each even degree 0 ≤ 2k ≤ 2n, ∗ with trivial boundary homomorphisms. Hence CCW (CP n ) has trivial cobound2k n ∼ ary, so H (CP ) = Z for each 0 ≤ k ≤ n, and the other cohomology groups are 0.

CHAPTER 3. CUP PRODUCT Proposition 3.7.2.

51

H ∗ (CP n ) ∼ = Z[y]/(y n+1 )

and

H ∗ (CP ∞ ) ∼ = Z[y] ,

where |y| = 2. Let HP n be the n-dimensional quaternionic projective space, of real dimenS sion 4n, and let HP ∞ = n HP n . The cellular complex C∗CW (HP n ) has one generator e4k in degree 4k, for 0 ≤ k ≤ n, and trivial boundary homomor∗ phisms. Hence CCW (HP n ) has trivial coboundary, so H 4k (HP n ) ∼ = Z for each 0 ≤ k ≤ n, and the other cohomology groups are 0. Proposition 3.7.3.

H ∗ (HP n ) ∼ = Z[z]/(z n+1 )

and

H ∗ (HP ∞ ) ∼ = Z[z] ,

where |z| = 2.

3.8

Hopf maps

One way to detect whether a map f : X → Y is null-homotopic or not is to consider the cup product structure in the cohomology of the mapping cone Cf = Y ∪ CX. f

X

/Y i

/ Cf

j r

If f is null-homotopic, then there is a retraction r : Cf → Y , so that the ring homomorphisms r∗

j∗

H ∗ (Y ) −→ H ∗ (Cf ) −→ H ∗ (Y ) split off H ∗ (Y ) as a graded subring of H ∗ (Cf ). Therefore, if H ∗ (Y ) does not split off from H ∗ (Cf ), then f cannot be null-homotopic, i.e., it must be an essential map. For example, in the CW structure on CP 2 , the 4-cell is attached to the 2-skeleton CP 1 = S 2 by the complex Hopf map η : S3 → S2 taking a point in S 3 ⊂ C2 to the complex line that goes though it. The mapping cone is Cη = CP 2 . Here H ∗ (CP 2 ) = Z[y]/(y 3 ) = Z{1, y, y 2 } restricts by j ∗ to H ∗ (S 2 ) = Z[y]/(y 2 ) = Z{1, y} , but since y 2 = 0 in H ∗ (S 2 ) and y 2 6= 0 in H ∗ (CP 2 ) there is no ring homomorphism r∗ : H ∗ (S 2 ) → H ∗ (CP 2 ) that would be a section to j ∗ . Hence η cannot be null-homotopic.

CHAPTER 3. CUP PRODUCT

52

As a similar example, in the CW structure on HP 2 , the 8-cell is attached to the 4-skeleton HP 1 = S H by the quaternionic Hopf map ν : S7 → S4 taking a point in S 7 ⊂ H2 to the quaternionic line that goes though it. The mapping cone is Cν = HP 2 . Here H ∗ (HP 2 ) = Z[z]/(z 3 ) = Z{1, z, z 2 } restricts by j ∗ to H ∗ (S 4 ) = Z[z]/(z 2 ) = Z{1, z} , but since z 2 = 0 in H ∗ (S 4 ) and z 2 6= 0 in H ∗ (HP 2 ) there is no ring homomorphism r∗ : H ∗ (S 4 ) → H ∗ (HP 2 ) that would be a section to j ∗ . Hence ν cannot be null-homotopic. There is also an octonionic plane, denoted OP 2 , with cohomology ring ∗ H (OP 2 ) ∼ = Z[w]/(w3 ) with |w| = 8, and the attaching map σ : S 15 → S 8 is an essential map known as the octonionic Hopf map. A more careful argument shows that η has infinite order in π3 (S 2 ), ν has infinite order in π7 (S 4 ) and σ has infinite order in π15 (S 8 ).

3.9

Graded commutativity

Recall that H ∗ (T 2 ) = Z{1, α, β, γ} with |α| = |β| = 1 and |γ| = 2, with α ∪ β = γ = −β ∪ α. This commutativity up to a sign is typical. Theorem 3.9.1. Let (X, A) be a pair of space and let R be a commutative ring. Then β ∪ α = (−1)k` α ∪ β in H k+` (X, A; R), for all α ∈ H k (X, A; R) and β ∈ H ` (X, A; R). We say that H ∗ (X, A; R) is graded commutative, or that it is a commutative graded ring. Note that if H ∗ (X, A; R) is concentrated in even degrees, then the sign (−1)k` is always +1. Proof. Let ρ : Cn (X) → Cn (X) be the (natural) chain map that takes σ : ∆n → X to ρ(σ) = n σ|[vn , . . . , v0 ] , where σ|[vn , . . . , v0 ] is the composite of the affine linear map ρn : ∆n → ∆n that reverses the ordering of the vertices, and n = (−1)n(n+1)/2 is the sign of the associated permutation. ((Check that ∂ρ = ρ∂, by calculation.)) There is a (natural) chain homotopy P : Cn (X) → Cn+1 (X) from the identity 1 to ρ. ((Define and check.))

CHAPTER 3. CUP PRODUCT

53

We get an induced chain map ρ∗ : C n (X; R) → C n (X; R) and chain homotopy P ∗ from 1 to ρ∗ . Recall the definition of the cochain level cup product of ϕ : Ck (X) → R and ψ : C` (X) → R: (ϕ ∪ ψ)(σ) = ϕ(σ|[v0 , . . . , vk ]) · ψ(σ|[vk , . . . , vn ]) for σ : ∆n → X as above, n = k + `. Then ρ∗ (ψ ∪ ϕ)(σ) = n (ψ ∪ ϕ)(σ|[vn , . . . , v0 ]) = n ψ(σ|[vn , . . . , vk ]) · ϕ(σ|[vk , . . . , v0 ]) while (ρ∗ ϕ ∪ ρ∗ ψ)(σ) = ϕ(k σ|[v0 , . . . , vk ]) · ψ(` σ|[vk , . . . , vn ]) = k ` ϕ(σ|[v0 , . . . , vk ]) · ψ(σ|[vk , . . . , vn ]) . Using the relation n = (−1)k` k ` and commutativity of R, we get that ρ∗ (ψ ∪ ϕ) = (−1)k` ρ∗ ϕ ∪ ρ∗ ψ . Hence, at the level of cohomology groups, β ∪ α = [ψ ∪ ϕ] = [ρ∗ (ψ ∪ ϕ)] = (−1)k` [ρ∗ ϕ ∪ ρ∗ ψ] = (−1)k` [ρ∗ ϕ] ∪ [ρ∗ ψ] = (−1)k` [ϕ] ∪ [ψ] = (−1)k` α ∪ β when ϕ and ψ are cocycles representing α and β.

3.10

Tensor products of graded rings

If A∗ and B∗ are graded rings, we define their tensor product A∗ ⊗ B∗ to be the tensor product of graded abelian groups, with M [A∗ ⊗ B∗ ]n = Ak ⊗ B` k+`=n

in degree n, with the graded multiplication [A∗ ⊗ B∗ ]n ⊗ [A∗ ⊗ B∗ ]n0 −→ [A∗ ⊗ B∗ ]n+n0 given by

0

(α ⊗ β) · (α0 ⊗ β 0 ) = (−1)k` αα0 ⊗ ββ 0 where |β| = k and |α0 | = `0 . In terms of diagrams, the multiplication on A∗ ⊗B∗ is the composite 1⊗τ ⊗1

µ⊗µ

A∗ ⊗ B∗ ⊗ A∗ ⊗ B∗ −→ A∗ ⊗ A∗ ⊗ B∗ ⊗ B∗ −→ A∗ ⊗ B∗ , where τ : B∗ ⊗A∗ → A∗ ⊗B∗ is the graded twist isomorphism that takes β ⊗α0 to 0 (−1)k` α0 ⊗ β, with notation as above, and µ : A∗ ⊗ A∗ → A∗ and µ : B∗ ⊗ B∗ → B∗ are the multiplications in A∗ and B∗ . ((Example with products of spheres?))

Chapter 4

K¨ unneth theorems 4.1

A K¨ unneth formula in cohomology

Let (X, A) and (Y, B) be pairs of spaces, and let R be a commutative ring. Recall the notation (X, A) × (Y, B) = (X × Y, A × Y ∪ X × B). Theorem 4.1.1 (K¨ unneth formula). The cross product ×

H ∗ (X, A; R) ⊗R H ∗ (Y, B; R) −→ H ∗ ((X, A) × (Y, B); R) is an isomorphisms of graded rings, if (X, A) and (Y, B) are pairs of CW complexes and H ` (Y, B; R) is a finitely generated projective R-module, for each `.

4.2

The K¨ unneth formula in homology

Let R be a PID, throughout this section. Theorem 4.2.1 (K¨ unneth formula). There is a natural short exact sequence 0→

M

×

Hk (X; R) ⊗R H` (Y ; R) −→ Hk+` (X × Y ; R) −→

k+`=n

M

−→

TorR 1 (Hk (X; R), H` (Y ; R)) → 0

k+`=n−1

for each n, and these sequences split. The hypothesis of the following consequence is automatic if R is a field. Corollary 4.2.2. Suppose that H` (Y ; R) is flat over R, for each `. There is a natural isomorphism ∼ =

× : H∗ (X; R) ⊗R H∗ (Y ; R) −→ H∗ (X × Y ; R) . One proof of the theorem goes in two parts. One is the Eilenberg–Zilber theorem, relating the chains on X × Y to the algebraic tensor product of the chains on X and the chains on Y . The other is the algebraic K¨ unneth theorem, computing the homology of a tensor product of chain complexes. 54

¨ CHAPTER 4. KUNNETH THEOREMS

55

((Reference to Spanier’s “Algebraic topology” or Mac Lane’s “Homology”.)) Here is the external version of the Alexander–Whitney diagonal approximation. Definition 4.2.3. The Alexander–Whitney homomorphism M AWn : Cn (X × Y ; R) −→ Ck (X; R) ⊗R C` (Y ; R) k+`=n n

takes (σ, τ ) : ∆ → X × Y to σ|[v0 , . . . , vk ] ⊗ τ |[vk , . . . , vn ] . Theorem 4.2.4 (Eilenberg–Zilber theorem). The Alexander–Whitney homomorphism is a chain homotopy equivalence '

AW# : C∗ (X × Y ; R) −→ C∗ (X; R) ⊗R C∗ (Y ; R) . To prove this, one can construct a chain homotopy inverse '

EZ# : C∗ (X; R) ⊗R C∗ (Y ; R) −→ C∗ (X × Y ; R) . This can either be done by the method of acyclic models, or by an explicit formula, known as the Eilenberg–Zilber shuffle homomorphism. ((ETC)) Theorem 4.2.5 (Algebraic K¨ unneth formula). Let (C∗ , ∂) and (D∗ , ∂) be chain complexes of free R-modules. Then there is a natural short exact sequence M 0→ Hk (C∗ ) ⊗R H` (D∗ ) −→ Hn (C∗ ⊗ D∗ ; R) −→ k+`=n

−→

M

TorR 1 (Hk (C∗ ), H` (D∗ )) → 0

k+`=n−1

for each n, and these sequences split. The proof is similar to that of the universal coefficient theorem. Under the assumption that Hk (X; R) and H` (Y ; R) are finitely generated projective R-modules, for each k and `, we can dualize the homological K¨ unneth isomorphism ∼ = H∗ (X; R) ⊗R H∗ (Y ; R) −→ H∗ (X × Y ; R) and use the universal coefficient theorem to get a cohomological K¨ unneth isomorphism H ∗ (X × Y ; R) ∼ = HomR (H∗ (X × Y ; R), R) ∼ HomR (H∗ (X; R) ⊗R H∗ (Y ; R), R) = ∼ = HomR (H∗ (X, R), R) ⊗R HomR (H∗ (Y ; R), R) ∼ = H ∗ (X; R) ⊗R H ∗ (Y ; R) . Notice how finite generation is needed in the middle, using that the homomorphism HomR (M, R) ⊗R HomR (N, R) −→ HomR (M ⊗R N, R) , taking the tensor product of ϕ : M → R and ψ : N → R to the composite ϕ⊗ψ

M ⊗R N −→ R ⊗R R ∼ = R, is an isomorphism when M and N are finitely generated projective R-modules.

¨ CHAPTER 4. KUNNETH THEOREMS

4.3

56

Proof of the cohomology K¨ unneth formula

We will instead give a different proof of the cohomological K¨ unneth isomorphism, based on the study of generalized cohomology theories, which leads more directly to a result with weaker hypotheses. The main part of the proof deals with the absolute case when B = ∅, saying that the cross product ×

H ∗ (X, A; R) ⊗R H ∗ (Y ; R) −→ H ∗ (X × Y, A × Y ; R) is an isomorphism of graded rings, if (X, A) is a pair of CW complexes and H ` (Y ; R) is a finitely generated projective R-module, for each `. The relative case, when (Y, B) is a pair of CW complexes and H ` (Y, B; R) is a finitely generated projective R-module, for each `, follows by naturality with respect to the map (Y, B) → (Y /B, B/B) inducing isomorphisms ∼ =

H ` (Y /B, B/B; R) −→ H ` (Y, B; R) for all `, and the splittings H ∗ (Y /B; R) ∼ = H ∗ (Y /B, B/B; R) ⊕ H ∗ (B/B; R) and H ∗ ((X, A) × Y /B; R) ∼ = H ∗ ((X, A) × (Y /B, B/B); R) ⊕ H ∗ ((X, A) × B/B; R) . Consider the functors of CW pairs (X, A) given by M hn (X, A) = H k (X, A; R) ⊗R H ` (Y ; R) k+`=n

and k n (X, A) = H n (X × Y, A × Y ; R) . The cross product defines a natural transformation µ : hn (X, A) −→ k n (X, A) for all n. We prove that h∗ and k ∗ are cohomology theories on the category of CW pairs, and that µ is a map of cohomology theories, i.e., a natural transformation that commutes with the connecting homomorphisms. Proposition 4.3.1. If a map µ : h∗ → k ∗ of cohomology theories on the category of CW pairs is an isomorphism on the pair (?, ∅), then it is an isomorphism for all CW pairs. Proof. By the map of long exact sequences hn−1 (X) µ

/ hn−1 (A)

δ

µ



k n−1 (X)

/ hn (X, A) µ



/ k n−1 (A)

δ

 / k n (X, A)

/ hn (X) µ

 / k n (X)

/ hn (A) µ

 / k n (A)

¨ CHAPTER 4. KUNNETH THEOREMS

57

and the five-lemma, it suffices to prove the proposition in the case when A = ∅. In this absolute case first we proceed by induction on` the dimension m of X. When X is 0-dimensional, it is the disjoint union X = α ? of a set of points, so by the commutative diagram ` h∗ ( α ?)

∼ =

/

Q

α

h∗ (?) Q

µ

` k ∗ ( α ?)

∼ =

/

Q

α

µ

 ∗ α k (?)

` and the hypothesis for X = ?, it follows that µ is an isomorphism for X = α ?. Let m ≥ 1, assume that µ is an isomorphism for all X of dimension less than m, and suppose that X = X (m) has dimension m. By the map of long exact sequences above in the case (X, A) = (X (m) , X (m−1) ), the inductive hypothesis and the five-lemma, it suffices to prove that µ is an isomorphism for this CW pair. Let a Φ: (Dm , ∂Dm ) → (X (m) , X (m−1) ) α

be the characteristic maps of the m-cells of X. In the commutative diagram h∗ (X (m) , X (m−1) )

Φ∗

µ

 k ∗ (X (m) , X (m−1) )

∼ =

` m m / h∗ ( α (D , ∂D ))

/

Q

α

µ Φ∗

h∗ (Dm , ∂Dm ) µ



∼ =

` m m / k∗ ( α (D , ∂D ))

/

Q

 ∗ m k (D , ∂Dm ) α

the homomorphisms labeled Φ∗ are isomorphisms by excision, and the righthand horizontal arrows are isomorphisms by the product axiom. Hence it suffices to prove that µ is an isomorphism for the CW pair (Dm , ∂Dm ). By the map of long exact sequences above, in the case (X, A) = (Dm , ∂Dm ), it suffices to know that µ is an isomorphism for X = Dm and for X = ∂Dm . The first follows from the case X = ? naturality with respect to the map Dm → ? and homotopy invariance. The second follows by induction, since the dimension of ∂Dm is less than m. The case of infinite-dimensional X remains. For this we use that X is the colimit of its skeleta, in the sense that there is a sequence of cellular inclusions of CW complexes [ X (m−1) ⊂ X (m) ⊂ · · · ⊂ X = X (m) . m

There is a mapping telescope [ T = [m, m + 1] × X (m) ⊂ R × X m

and the composite projection T ⊂ R × X → X is a homotopy equivalence. See [2, Lemma 2.34]. We can write this mapping telescope as the homotopy coequalizer of two maps a a 1, i : X (m) −→ X (m) m

m

¨ CHAPTER 4. KUNNETH THEOREMS

58

where 1 is the coproduct of the identity maps X (m) → X (m) , while i is the coproduct of the inclusion maps X (m−1) → X (m) . Hence there is a natural long exact sequence a a a a 1−i∗ 1−i∗ hn−1 ( X (m) ) −→ hn−1 ( X (m) ) −→ hn (T ) −→ hn ( X (m) ) −→ hn ( X (m) ) m

m

m

m

which we can rewrite, using the product axiom and the homotopy equivalence T ' X, as Y

1−i∗

hn−1 (X (m) ) −→

m

Y

hn−1 (X (m) ) −→ hn (X) −→

m

Y

1−i∗

hn (X (m) ) −→

m

Y

hn (X (m) )

m

The kernel of the right hand 1 − i∗ consists of the compatible sequences (xm )m with xm ∈ hn (X (m) ) and i∗ (xm ) = xm−1 for all m, i.e., it equals the limit group ker(1 − i∗ ) = lim hn (X (m) ) . m

By definition, the cokernel of the left hand 1 − i∗ is the derived limit group cok(1 − i∗ ) = Rlim hn−1 (X (m) ) . m

It vanishes if the homomorphisms i∗ : hn (X (m) → hn (X (m−1) are surjective for sufficiently large m. These considerations are natural in the cohomology theory h, so there is a map of short exact sequences 0

/ Rlimm hn−1 (X (m) ) µ

µ



0

/ hn (X)

/ Rlimm k n−1 (X (m) )

 / k n (X)

/ limm hn (X (m) )

/0

µ

 / limm k n (X (m) )

/0

We have already shown that µ is an isomorphism for each finite-dimensional X (m) , hence is induces an isomorphism of limits and derived limits. Thus µ is also an isomorphism for the general CW complex X. Proof of the cohomology K¨ unneth formula. We must exhibit h∗ and k ∗ as cohomology theories, check that µ is a map of such, and that µ is an isomorphism for the one-point space ?. The connecting homomorphism δ : hn−1 (A) → hn (X, A) is defined as the direct sum of the tensor products δ ⊗ 1 : H k−1 (A; R) ⊗R H ` (Y ; R) −→ H k (X, A; R) ⊗R H ` (Y ; R) as k ranges over the integers and ` = n − k. The connecting homomorphism δ : k n−1 (A) → k n (X, A) is the usual connecting homomorphism δ : H n−1 (A × Y ; R) −→ H n (X × Y, A × Y ; R) of the pair (X × Y, A × Y ).

¨ CHAPTER 4. KUNNETH THEOREMS

59

The tensor product of the long exact sequence i∗

j∗

δ

i∗

H k−1 (X; R) −→ H k−1 (A; R) −→ H k (X, A; R) −→ H k (X; R) −→ H k (A; R) with H ` (Y ; R) over R is still exact, because H ` (Y ; R) is projective, hence flat. Summing over all k + ` = n we get the long exact sequence for h∗ . The long exact sequence for k ∗ at (X, A) is just the usual long exact sequence for H n (−; R) at (X × Y, A × Y ). Homotopy invariance for h∗ and k ∗ follows immediately from homotopy invariance for ordinary cohomology. Excision, either in the form for general topological pairs, or in the form for subcomplexes of a CW complex, is also obvious for h∗ . The case of k ∗ is about as easy, since if Z ⊆ A ⊆ X with the closure of Z contained in the interior of A, the Z × Y ⊂ A × Y ⊂ X × Y with the closure of Z × Y contained in the interior of A × Y , and similarly for products of subcomplexes of X with Y . ` ∗ The product axiom is clear for k , since if X = X α α then X × Y = ` (X × Y ), and similarly in the relative case. The product axiom for h∗ is α α more subtle. It amounts to the assertion that Y Y ( H k (Xα , Aα ; R)) ⊗R H ` (Y ; R) −→ (H k (Xα , Aα ; R) ⊗R H ` (Y ; R)) α

α

is an isomorphism, for all k and `. This is clear if H ` (Y ; R) = R, hence also if H ` (Y ; R) is finitely generated and free, since finite sums of R-modules are also finite products. By naturality in H ` (Y ; R), it also follows when H ` (Y ; R) is finitely generated and projective. The assertion that µ is a map of generalized cohomology theories is clear from the naturality of the cross product, together with the previously proved formula δ(α × η) = δα × η, relating the connecting homomorphism to the cross product. The assertion that µ is an isomorphism for (X, A) = (?, ∅) is the assertion that × : H 0 (?; R) ⊗R H ` (Y ; R) −→ H ` (? × Y ; R) is an isomorphism for all `, which is clear. Theorem 4.3.2 (Hopf ). If there is a real division algebra structure on Rn then n is a power of 2. Proof. A division algebra structure on Rn is a bilinear pairing · : Rn × Rn → Rn such that x · y = 0 only if x = 0 or y = 0. Given such a pairing, we have a map g : S n−1 × S n−1 → S n−1 given by g(x, y) = x · y/|x · y|, such that g(−x, y) = −g(x, y) = g(x, −y) . Passing to quotients we get a map h : RP n−1 × RP n−1 → RP n−1 . We may assume that n > 2, in which case π1 (RP n−1 ) ∼ = Z/2. The displayed formula, and a consideration of covering spaces, implies that h induces the sum homomorphism on π1 (RP n−1 ). Passing to cohomology, we have a graded ring isomorphism H ∗ (RP n−1 ; Z/2) ∼ = Z/2[γ]/(γ n = 0) ,

¨ CHAPTER 4. KUNNETH THEOREMS

60

with deg(γ) = 1, and by the K¨ unneth formula, H ∗ (RP n−1 × RP n−1 ; Z/2) ∼ = Z/2[α, β]/(αn = 0, β n = 0) The formula for h on π1 implies that h∗ (γ) = α + β in H 1 , so 0 = h∗ (γ n ) = (α + β)n =

n−1 X i=1

in H n . It is a number-theoretic fact that and) only if n is a power of 2.


n i

 n i n−i αβ i

≡ 0 mod 2 for all 0 < i < n (if

Chapter 5

Poincar´ e duality Definition 5.0.3. An n-dimensional manifold is a Hausdorff space M such that each point has an open neighborhood that is homeomorphic to Rn . If M is also compact as a topological space, then we call M a closed n-manifold. Poincar´e duality asserts that for a closed, orientable n-manifold M there is an isomorphism Hk (M ; Z) ∼ = H n−k (M ; Z) for each integer k. Without the orientability hypothesis, there is an isomorphism Hk (M ; Z/2) ∼ = H n−k (M ; Z/2) for each k.

5.1

Orientations

Let R be any commutative ring. Let x ∈ M be any point, and let U ⊆ M be an open neighborhood of x with (U, x) ∼ = (Rn , 0). Then there are isomorphisms Hk (M, M − {x}; R) ∼ = Hk (U, U − {x}; R) ∼ = Hk (Rn , Rn − {0}; R) ∼ e k−1 (Rn − {0}; R) =H for each integer k. Hence ( R Hk (M, M − {x}; R) ∼ = 0

if k = n otherwise

It follows that the dimension n is an intrinsic property of the manifold M (when nonempty). Definition 5.1.1. By a local orientation of M at x we mean a choice µx ∈ Hn (M, M −{x}; Z) of one of the two possible generators of Hn (M, M −{x}; Z) ∼ = Z, as an abelian group. Under the isomorphisms above, this corresponds to a choice of a generator of Hn (Rn , Rn − {0}; Z), which in turn can be identified with a choice of vector space basis for Rn , up to the equivalence relation that 61

´ DUALITY CHAPTER 5. POINCARE

62

considers two bases to be equivalent if the change-of-basis matrix has positive determinant. More generally, by a local R-orientation of M at x we mean a choice of a generator µx of Hn (M, M − {x}; R) ∼ = R, as an R-module. When R = Z/2 there is one one possible such choice, namely the nonzero element in Hn (M, M − {x}; R). A local orientation is thus the same as a local Z-orientation. Definition 5.1.2. The following notational convention is helpful: For A ⊆ X we let Hk (X|A) = Hk (X, X − A) for all integers k, and similarly with arbitrary coefficient groups. We may read X|A as X relative to the complement of A. In the special case when A = {y} we simply write Hk (X|y) for Hk (X|{y}). If A ⊆ B ⊆ X the inclusion X − B ⊆ X − A induces homomorphisms Hk (X|B) −→ Hk (X|A) which we say are given by restriction along A ⊆ B. If U is a neighborhood of the closure of A in X, then the inclusion U ⊆ X induces isomorphisms ∼ =

Hk (U |A) −→ Hk (X|A) by excision, so H∗ (X|A) only depends on this neighborhood. We may therefore call Hk (X|A) the local homology of X at A. A local R-orientation µx of an n-manifold M at a point x determines a local R-orientation µy of M at all points y in a neighborhood of x, by the following prescription: Let U ⊆ M be an open neighborhood of x with (U, x) ∼ = (Rn , 0), and let B ⊂ U be a smaller open neighborhood that corresponds to an open ball B n (r) ⊂ Rn of finite radius r, centered at the origin. Then the inclusion U − B ⊂ U − {y} is a homotopy equivalence for each point y ∈ B, so the restriction map ∼ = Hn (M |B; R) −→ Hn (M |y; R) is an isomorphism, for each point y ∈ B. Let µB ∈ Hn (M |B; R) be the preimage of the generator µx ∈ Hn (M |x; R), in the special case y = x, and let µy ∈ Hn (M |y; R) be the image of µB , for all y ∈ B. Then each µy is a local Rorientation of M at y. These local R-orientations at points y 6= x may depend on the choice made of the neighborhood U ∼ = Rn . If it is possible to make a consistent choice of local R-orientations, at all points of M , we say that M is R-orientable: Definition 5.1.3. An R-orientation of an n-manifold M is a choice of a local R-orientation µx ∈ Hn (M |x; R) at each point x ∈ M , such that for each x ∈ M there are open neighborhoods B ⊂ U with (U, B, x) ∼ = (Rn , B n (r), 0), and the class µB ∈ Hn (M |B; R) that restricts to µx ∈ Hn (M |x; R) restricts to µy ∈ Hn (M |y; R) for each y ∈ B. An orientation of M is the same as a Z-orientation. Any n-manifold M has a unique Z/2-orientation, with µx ∈ Hn (M |x; Z/2) ∼ = Z/2 equal to the nonzero element for each x ∈ M . This is clear, since the restriction isomorphisms

´ DUALITY CHAPTER 5. POINCARE

63

∼ Hn (M |y; Z/2) always take the (unique) nonzero element to Hn (M |B; Z/2) = the nonzero element. The consistency condition on local R-orientations at points can be re-expressed as a continuity condition for a section in a covering space π : MR → M called the R-orientation covering of M . Definition 5.1.4. Let M be an n-manifold and let R be any commutative ring. Let a MR = Hn (M |x; R) x∈M

be the disjoint union of the local homology groups Hn (M |x; R) = Hn (M, M − {x}; R), for all points x ∈ M . For each pair of open neighborhoods B ⊂ U ⊆ M , with (U, B) ∼ = (Rn , B n (r)) for some finite r, and each class α ∈ Hn (M |B; R), let V (B, α) ⊆ MR be the set of elements αx ∈ Hn (M |x; R) ⊆ MR with x ∈ B and αx the restriction of α. The collection of subsets V (B, α) for all such B and α is a basis for a topology on MR , making MR a topological space. The map π : MR → M taking all of Hn (M |x; R) ⊆ MR to x ∈ M is a covering space projection. For each pair B ⊂ U as above, the preimage π −1 (B) decomposes as π −1 (B) ∼ = B × Hn (M |B; R) over B, where Hn (M |B; R) ∼ = R has the discrete topology. The isomorphism takes (x, α) for x ∈ B, α ∈ Hn (M |B; R) to the restriction µx ∈ Hn (M |x; R) ⊂ π −1 (B) of α along {x} ⊂ B. Each fiber π −1 (x) = Hn (M |x; R) is isomorphic to R, as an R-module. An R-orientation of M is then the same as a continuous section µ : M → MR (with π ◦ µ = 1M ), taking x ∈ M to an R-module generator µx ∈ π −1 (x) for each x. Definition 5.1.5. Let ΓR (M ) be the set of (continuous) sections α : M → MR , mapping x ∈ M to αx ∈ MR . It is an R-module, with (r · α)x = r · αx in Hn (M |x; R), for r ∈ R and x ∈ M . Theorem 5.1.6. Let M be a closed, connected n-manifold. (a) Hk (M ; R) = 0 for k > n. (b) If M is R-orientable, then the restriction map ∼ = Hn (M ; R) −→ Hn (M |x; R) ∼ =R

is an isomorphism, for each x ∈ X. Definition 5.1.7. An element [M ] ∈ Hn (M ; R) whose image in Hn (M |x; R) is a generator for all x ∈ M is called a fundamental class or an orientation class for M , with coefficients in R. Specifying a fundamental class for M is equivalent to specifying an R-orientation for M . The theorem follows from the following more precise statement. Lemma 5.1.8. Let M be an n-manifold and let A ⊆ M be a compact subset.

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64

(a) Hk (M |A; R) = 0 for k > n. (b) If α ∈ ΓR (M ) is a section of the covering space MR → M , then there exists a unique class αA ∈ Hn (M |A; R) whose restriction to Hn (M |x; R) is αx for each x ∈ A. In other words, part (b) of the lemma says that the natural homomorphism Hn (M |A; R) −→ {lifts A → MR of A → M } is injective, and that it image contains the image of the homomorphism ΓR (M ) −→ {lifts A → MR of A → M } that takes a section α : M → MR to its restriction α|A. Proof of theorem. When M is compact, the case A = M of the lemma says that Hk (M ; R) = 0 for k > n, and that there is an isomorphism Hn (M ; R) ∼ = ΓR (M ) . If M is R-orientable, we can choose an R-orientation x 7→ µx , which is a section µ ∈ ΓR (M ) such that µx is a generator of Hn (M |x; R) for each x ∈ M . Hence any section α ∈ ΓM (R) can be uniquely written as ((ETC)) Proof of lemma. We omit R from the notation. The proof goes in four steps. Step (1): If the lemma is true for compact subsets A, B and A ∩ B of M , then it is true for A ∪ B. There is a long exact Mayer–Vietoris sequence ∂

Φ

Ψ

∂

. . . −→ Hk (M |A ∪ B) −→ Hk (M |A) ⊕ Hk (M |B) −→ Hk (M |A ∩ B) −→ . . . associated to the covering of M by the open subsets M − A and M − B. Here Φ takes α ∈ Hk (M |A ∪ B) to (α, α) in Hk (M |A) ⊕ Hk (M |B), and Ψ takes (α, β) ∈ Hk (M |A) ⊕ Hk (M |B) to α − β ∈ Hk (M |A ∩ B) (omitting notation for the restriction maps). Part (a) of the lemma follows by exactness, since each group Hk (M |A ∪ B) sits between two trivial groups, for k > n. For k = n, we have the half-exact sequence Φ

Ψ

0 → Hn (M |A ∪ B) −→ Hn (M |A) ⊕ Hn (M |B) −→ Hn (M |A ∩ B) . Suppose that x 7→ αx is a section in ΓR (M ). By assumption there are unique classes αA ∈ Hn (M |A), αB ∈ Hn (M |B) and αA∩B ∈ Hn (M |A ∩ B) such that αA restricts to αx for each x ∈ A, αB restricts to αy for each y ∈ B, and αA∩B restricts to αz for each z ∈ A ∩ B. By the uniqueness of αA∩B , both αA and αB restrict to αA∩B in Hn (M |A ∩ B). Hence Ψ(αA , αB ) = αA∩B − αA∩B = 0. By exactness and injectivity of Φ, it follows that there is a unique class αA∪B ∈ Hn (M |A ∪ B) that restricts to αA in Hn (M |A) and to αB ∈ Hn (M |B). This is equivalent to saying that it is the unique class that restricts to αx for all x ∈ A and to αy for all y ∈ B. This, in turn, is equivalent to part (b) of the lemma for A ∪ B. Step (2): It suffices to prove the lemma for M = Rn . Step (3): Step (4): ((ETC))

´ DUALITY CHAPTER 5. POINCARE

5.2

65

Cap product

Definition 5.2.1. Let X be a space and R a commutative ring. The cap product ∩ Ck+` (X; R) ⊗R C k (X; R) −→ C` (X; R) sends a (k + `)-simplex σ : ∆k+` → X and a k-cochain ϕ ∈ C k (X; R) to the `-chain σ ∩ ϕ = ϕ(σ|[v0 , . . . , vk ])σ|[vk , . . . , vk+` ] . Lemma 5.2.2. ∂σ ∩ ϕ = σ ∩ δϕ + (−1)k ∂(σ ∩ ϕ). Hence there is an induced cap product ∩

Hk+` (X; R) ⊗R H k (X; R) −→ H` (X; R) . For pairs (X, A) there are relative forms ∩

Hk+` (X, A; R) ⊗R H k (X; R) −→ H` (X, A; R) ∩

Hk+` (X, A; R) ⊗R H k (X, A; R) −→ H` (X; R) and, more generally, ∩

Hk+` (X, A + B; R) ⊗R H k (X, A; R) −→ H` (X, B; R) for subsets A, B ⊂ X. If A and B are excisive in X, the source can be rewritten as Hk+` (X, A + B; R) ∼ = Hk+` (X, A ∪ B; R) . Lemma 5.2.3 (Projection formula). Let f : X → Y be any map. Then f∗ (α) ∩ ϕ = f∗ (α ∩ f ∗ (ϕ)) in C` (Y ; R) for α ∈ Ck+` (X; R) and ϕ ∈ C k (Y ; R), hence also in H` (Y ; R) for α ∈ Hk+` (X; R) and ϕ ∈ H k (Y ; R). In other words, f∗ : H∗ (X; R) → H∗ (Y ; R) is an H ∗ (Y ; R)-module homomorphism, where H ∗ (Y ; R) acts on H∗ (Y ; R) by cap product, and on H∗ (X; R) via the ring homomorphism f ∗ : H ∗ (Y ; R) → H ∗ (X; R) followed by cap product. Theorem 5.2.4 (Poincar´ e duality). Let M be a closed, R-oriented n-manifold with fundamental class [M ] ∈ Hn (M ; R). Then the homomorphism DM : H k (M ; R) −→ Hn−k (M ; R) , given by DM (ϕ) = [M ] ∩ ϕ, is an isomorphism for all k.

5.3

Cohomology with compact supports

Fix a space X. The partially ordered set of compact subsets A ⊂ X is directed, in the sense that it is nonempty, and any two compact subsets A, B ⊂ X are contained in a larger compact subset, such as their union A ∪ B.

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66

For each compact A, there is a subcomplex C ∗ (X|A) = C ∗ (X, X − A) ⊂ C ∗ (X) of cochains that vanish on simplices in the complement of A. If A ⊂ B then restriction along i : (X, X −B) ⊂ (X, X −A) induces an inclusion i∗ : C ∗ (X|A) ⊂ C ∗ (X|B) of subcomplexes. The union of all of these subcomplexes, the colimit [ Cc∗ (X) = C ∗ (X|A) = colim C ∗ (X|A) , A compact

A compact

is the subcomplex of compactly supported cochains in C ∗ (X). (Co-)homology commutes with directed colimits, so Hc∗ (X) = H∗ (Cc∗ (X), δ) ∼ =

colim H ∗ (X|A)

A compact

defines the cohomology with compact supports of X. There is a natural homomorphism Hc∗ (X) → H ∗ (X), which is an isomorphism for X compact. Similarly with coefficients in any abelian group G. Lemma 5.3.1. Hck (Rn ; G)

( G ∼ = 0

for k = n otherwise.

¯ n (r) of Proof. Each compact subset A ⊂ Rn is contained in a closed ball B positive radius, centered at the origin. We say that these balls form a cofinal family within the directed set of compact subsets. Hence ¯ n (r); G) . Hc∗ (Rn ; G) ∼ = colim H ∗ (Rn |B r

k

n

¯n

Here H (R |B (r); G) is G for k = n and 0 for k 6= n, since '

¯ n (r)) −→ (Rn , Rn − {0}) (Rn , Rn − B is a homotopy equivalence for all positive r. Furthermore, the restriction homomorphisms for varying r are all isomorphisms, so the colimit system is constant, with colimit G for k = n and 0 otherwise.

5.4

Duality for noncompact manifolds

Definition 5.4.1. Let M be an R-oriented n-manifold, possibly noncompact. For each compact A ⊂ M there is a unique element µA ∈ Hn (M |A; R) that restricts to the given orientation of M at all points x ∈ A. Cap product defines a homomorphism µA ∩ (−) : H k (M |A; R) −→ Hn−k (M ; R) taking ϕ ∈ H k (M |A; R) to µA ∩ϕ ∈ Hn−k (M ; R). For compact A ⊂ B ⊂ M , the identity µA ∩ ϕ = µB ∩ i∗ (ϕ) holds, since i∗ (µB ) = µA , so these homomorphism combine to define a homomorphism DM : Hck (M ; R) −→ Hn−k (M ; R) for each integer k.

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Theorem 5.4.2. Let M be an R-oriented n-manifold. Then the homomorphism DM : Hck (M ; R) −→ Hn−k (M ; R) that extends µA ∩ (−) on H k (M |A; R) for each compact A ⊂ M , is an isomorphism for each integer k. This results contains the Poincar´e duality theorem for closed manifolds, in the case when M is compact. Lemma 5.4.3. Suppose that M = U ∪ V is a union of two open subsets. Then there is a diagram of Mayer–Vietoris sequences Hck (U ∩ V ) DU ∩V

 Hn−k (U ∩ V )

/ H k (M ) c

/ H k (U ) ⊕ H k (V ) c c DU ⊕DV

DM



/ Hn−k (U ) ⊕ H k (V ) c

/ H k+1 (U ∩ V ) c DU ∩V



 / Hn−k−1 (U ∩ V )

/ Hn−k (M )

that commutes up to sign. Proof. Let A ⊂ U and B ⊂ V be compact. Then A ∩ B ⊂ U ∩ V is also compact. There are excision isomorphisms H k (M |A) ∼ = H k (U |A), etc. We get a diagram, where the upper row is the Mayer–Vietoris sequence in cohomology for (M, M − A) and (M, M − B), and the lower row is the Mayer–Vietoris sequence in homology for U and V . H k (M |A ∩ B) ∼ =

 H k (U ∩ V |A ∩ B) µA∩B

 Hn−k (U ∩ V )

/ H k (M |A) ⊕ H k (M |B)

/ H k (M |A ∪ B)

∼ =

 H k (U |A) ⊕ H k (V |B) µA ∩(−)⊕µB ∩(−)

/ H k+1 (M |A ∩ B) ∼ =

 H k+1 (U ∩ V |A ∩ B)

µA∪B

µA∩B



/ Hn−k (U ) ⊕ H k (V )



/ Hn−k (M )

 / Hn−k−1 (U ∩ V )

This is straightforward the check for the left hand and middle square, less so for the right hand square. We refer to [2, pp. 246–247] for the details. As A and B range over all compact subsets of U and V , respectively, the intersection A∩B ranges over all compact subsets of U ∩V , and the union A∪B ranges over a cofinal family of compact subsets of M . Hence the colimit over the directed set of all such A and B, of the diagrams above, is the diagram of the lemma, with exact rows and squares that commute up to sign. Proof of the duality theorem. Step (1): If M = U ∪ V with U and V open, and the theorem holds for U , V and U ∩ V , then it holds for M . This is clear by the previous lemma and S the 5-lemma. Step (2): If M = i Ui is the union of an increasing sequence of open subsets, and the theorem holds for each Ui , then it holds for M . Each compact subset of M is contained in some Ui , so colimi Hc∗ (Ui ) ∼ = Hc∗ (M ). Similarly, each simplex in M factors through some Ui , so colimi H∗ (Ui ) ∼ = H∗ (M ). The homomorphism ∼ =

DM : Hck (M ) −→ Hn−k (M )

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is then the colimit of the assumed isomorphisms ∼ =

DUi : Hck (Ui ) −→ Hn−k (Ui ) , hence is an isomorphism. Step (3): The theorem holds for M = Rn . This is clear from the lemma ¯ n (r) the cap computing Hc∗ (Rn ; G) for G = R, since for each closed ball B = B product Hn (Rn |B; R) × H n (Rn |B; R) −→ H0 (Rn ; R) takes the orientation class µB and a generator of H n (Rn |B; R) to a unit times a point in Rn , which represents a generator of H0 (Rn ; R). Step (4): The theorem holds for M an open subset of Rn . ((Write M as a countable union of convex open subsets. Induction over the number of convex subsets, and passage to a colimit.)) Step (5): The theorem holds for M a countable union of open subsets homeomorphic to Rn . ((ETC))

5.5

Connection with cup product

Lemma 5.5.1. Let X be a space and R a commutative ring. The relation ψ(α ∩ ϕ) = (ϕ ∪ ψ)(α) holds for ϕ ∈ C k (X; R), ψ ∈ C ` (X; R) and α ∈ Ck+` (X; R). Hence the diagram H ` (X; R)

β

/ HomR (H` (X; R), R)

ϕ∪(−)

 H k+` (X; R)

β



((−)∩ϕ)∗

/ HomR (Hk+` (X; R), R)

commutes. An R-bilinear pairing A × B → R is said to be nonsingular if both of the induced homomorphisms A → HomR (B, R) and B → HomR (A, R) are isomorphisms. Lemma 5.5.2. Let M be an R-orientable n-manifold, with fundamental class [M ] ∈ Hn (M ; R). The cup pairing H k (M ; R) × H n−k (M ; R) −→ R taking (ϕ, ψ) to (ϕ ∪ ψ)[M ] is nonsingular when R is a field. When R = Z the induced pairing H k (M )/(torsion) × H n−k (M )/(torsion) −→ Z (of torsion-free quotient groups) is nonsingular.

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Lemma 5.5.3. Given an ordered basis α1 , . . . , αr for H k (M ; R) when R is a field (resp. for H k (M )/torsion for R = Z), there is a unique dual basis β1 , . . . , βr for H n−k (M ; R) (resp. for H n−k (M )/torsion), such that (αi ∪ βj )[M ] = δij is equal to 1 for i = j and 0 for i 6= j. In the middle-dimensional case n = 2k, the r × r matrix with (i, j)-th entry (αi ∪ αj )[M ] is an invertible matrix, hence has nonzero determinant when R is a field, and determinant ±1 when R = Z. It is symmetric if k is even, so that n ≡ 0 mod 4, and it is skew-symmetric if k is odd, so that n ≡ 2 mod 4. Remark 5.5.4. The classification of (skew-)symmetric integer matrices with determinant ±1 is interesting. These are known as (skew-)symmetric unimodular forms on Zr . A change of basis for Zr changes the integer matrix by conjugation, which gives an isomorphic unimodular form. In the skew-symmetric case, each unimodular matrix is conjugate to a block sum of copies of the matrix   0 −1 1 0 (realized by H 1 (T 2 )). In particular, the rank r must be even. In the symmetric case, the simplest examples are   1 (realized by H 2 (CP 2 )) and the form of  2 1 0 1 1 2 1 0  0 1 2 1  1 0 1 2  0 0 0 1  0 0 0 0  0 0 0 0 0 0 0 0

rank 8 known as E8  0 0 0 0 0 0 0 0  0 0 0 0  1 0 0 0  2 1 0 0  1 2 1 0  0 1 2 1 0 0 1 2

(which cannot be realized as H 2 (M ) for a closed, simply-connected smooth 4-manifold M , by a theorem of Rochlin). Corollary 5.5.5.

H ∗ (CP n ) ∼ = Z[y]/(y n+1 = 0) for n ≥ 0, and H ∗ (CP ∞ ) ∼ = Z[y], where deg(y) = 2. Proof. This is clear for n ≤ 1. Consider n ≥ 2 and assume that the result holds for CP n−1 . The inclusion CP n−1 → CP n induces an isomorphism in cohomology except in degree 2n, so if y ∈ H 2 (CP n ) is a generator, then y k ∈ H 2k (CP n ) is a generator for all 0 ≤ k < n. It remains to prove that it is also a generator for k = n. But this follows from Poincar´e duality for the orientable 2n-manifold CP n , since the pairing H 2 (CP n ) × H 2n−2 (CP n ) → Z

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is nonsingular, hence must take (y, y n−1 ) to a generator of Z. This means that y n evaluates to ±1 on the fundamental class of CP n , hence is a generator of H 2n (CP n ). The inclusion CP n → CP ∞ induces an isomorphism in cohomology in degrees ∗ ≤ 2n + 1, which proves the formula for H ∗ (CP ∞ ) in that range of degrees. Let n grow to infinity. Corollary 5.5.6. H ∗ (RP n ; Z/2) ∼ = Z/2[x]/(xn+1 = 0) for n ≥ 0, and H ∗ (RP ∞ ; Z/2) ∼ = Z/2[x], where deg(x) = 1. Proof. Same proof as for CP n , using Poincar´e duality for the Z/2-orientable n-manifold RP n . Let m ≥ 2 be a natural number. Let Cm ⊂ S 1 be the cyclic group of order m. Consider S 2n+1 to be the unit sphere in Cn+1 . The diagonal action by the complex numbers on Cn+1 restricts to a free action of S 1 on S 2n+1 , which further restricts to a free action of Cm on S 2n+1 . Let the lens space L2n+1 = S 2n+1 /Cm m be the orbit space. There is a CW structure on S 2n+1 with (2k + 1)-skeleton equal to S 2k+1 , and with 2k-skeleton equal to the join of S 2k−1 and Cm inside of S 2k−1 ∗ S 1 ∼ = S 2k+1 for each 0 ≤ k ≤ n. This structure has exactly m cells of each dimension from 0 to 2n + 1, and the Cm -action permutes the cells freely. Hence the orbit space has a CW structure with exactly one cell of each dimension from 0 to L2n+1 m , for each 0 ≤ k ≤ n. 2n + 1, with (2k + 1)-skeleton equal to L2k+1 m Corollary 5.5.7. Let p be an odd prime. H ∗ (L2n+1 ; Z/p) ∼ = Z/p[x, y]/(x2 = 0, y n+1 = 0) p 2 ∼ for n ≥ 0, and H ∗ (L∞ p ; Z/p) = Z/p[x, y]/(x ), where deg(x) = 1 and deg(y) = 2.

Proof. We omit the subscript p. The result is clear for n = 0, when L1 = S 1 /Cp is a circle, with cohomology H ∗ (L1 ; Z/p) ∼ = Z/p{1, x}. Let n ≥ 1 and suppose that the formula holds for L2n−1 . The inclusion L2n−1 → L2n+1 induces an isomorphism in cohomology with Z/p-coefficients, except in degrees 2n and 2n + 1, so if x ∈ H 1 (L2n+1 ; Z/p) and y ∈ H 2 (L2n+1 ; Z/p) are generators then y k ∈ H 2k (L2n+1 ; Z/p) and xy k ∈ H 2k+1 (L2n+1 ; Z/p) are generators for all 0 ≤ k < n. We have x2 = 0 by graded commutativity, since p is odd. It remains to prove that y n and xy n are generators. The second claim follows from Poincar´e duality for the Z/p-orientable (2n + 1)-manifold L2n+1 , since the pairing H 2 (L2n+1 ; Z/p) × H 2n−1 (L2n+1 ; Z/p) −→ Z/p is nonsingular, hence must take (y, xy n−1 ) to a generator. This means that y · xy n−1 = xy n is a generator. The first claim follows by writing this generator as the product x · y n .

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The cup product structure in H ∗ (L∞ m ; Z/m) is a little more complicated for non-prime m. Since S ∞ is contractible, the spaces CP ∞ , RP ∞ and L∞ p are classifying spaces for the groups S 1 , C2 and Cp , respectively. They are representatives for the homotopy types denoted BS 1 = BU (1), BC2 = BO(1) and BCp , and the corollaries above compute the cohomology algebras of the spaces of these homotopy types.

5.6

Other forms of duality

Let Rn+ ⊆ Rn be the half-space of points (x1 , . . . , xn ) where xn ≥ 0. An nmanifold with boundary is a Hausdorff space M such that each point has an open neighborhood that is homeomorphic to Rn or to Rn+ . For points x ∈ M that correspond to a point in Rn+ with xn = 0 (which we may assume is the origin), the local homology group Hn (M |x) = Hn (M, M − {x}) ∼ = Hn (Rn+ , Rn+ − {0}) ∼ =0 is trivial, unlike the local homology groups for x ∈ M that correspond to points in Rn+ with xn > 0 (or to points in Rn ). Hence the former points constitute a well-defined subspace ∂M ⊂ M , called the boundary of M . The subspace ∂M is an (n − 1)-dimensional manifold (without boundary). We say that a compact manifold M with boundary is R-orientable if M −∂M is orientable as a manifold without boundary. ((Explain fundamental class in Hn (M, ∂M ).)) Theorem 5.6.1 (Lefschetz duality). Let M be a compact R-oriented nmanifold with boundary, and suppose that ∂M = A ∪ B is the union of two compact (n−1)-manifolds A and B, with common boundary ∂A = ∂B. Then cap product with the fundamental class [M ] ∈ Hn (M, ∂M ; R) gives an isomorphism ∼ =

DM : H k (M, A; R) −→ Hn−k (M, B; R) for all k. ((Deduce for A = ∂M from Poincar´e duality, etc.))

Chapter 6

Vector bundles and classifying spaces We now follow Milnor and Stasheff’s book “Characteristic Classes” [3].

6.1

Real vector bundles

Definition 6.1.1. A family ξ of (real) vector spaces is a projection map π: E → B from the total space E = E(ξ) to the base space B = B(ξ), together with the structure of a (real) vector space on the fiber Fb = Fb (ξ) = π −1 (b), for each b ∈ B. If each vector space Fb has dimension n we say that ξ is a family of n-dimensional vector spaces. Example 6.1.2. Let B be any space. The trivial family nB of n-dimensional vector spaces is the projection π : B × Rn → B taking (b, x) to b for x ∈ Rn . Definition 6.1.3. A map ξ → η of families of vector spaces is a pair of maps g : E(ξ) → E(η) and g¯ : B(ξ) → B(η), such that E(ξ)

g

π

 B(ξ)

/ E(η) π

g ¯

 / B(η)

commutes and g restricts to a linear homomorphism Fb (ξ) → Fg¯(b) (η) for each b ∈ B(ξ). If ξ and η have the same base space B = B(ξ) = B(η), then a map ξ → η over B is a map of families (g, g¯) such that g¯ = idB .

72

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Lemma 6.1.4. A map (g, g¯) of families of vector spaces is an isomorphism if and only if g and g¯ are homeomorphisms. In particular, an isomorphism ξ ∼ = η over B is a homeomorphism g : E(ξ) → E(η) such that g / E(η) E(ξ) ∼ = CC { CC { { CC { { π C C! }{{{ π B commutes and g restricts to a linear isomorphism Fb (ξ) ∼ = Fb (η) for each b ∈ B. Definition 6.1.5. A trivialization of a family ξ of vector spaces is an isomorphism nB ∼ = ξ over B = B(ξ). More explicitly, this is a homeomorphism h : B × Rn → E(ξ) such that x 7→ h(b, x) is a linear isomorphism from Rn to Fb (ξ), for each b ∈ B. Definition 6.1.6. If π : E → B is a family of vector spaces ξ, and A ⊂ B is a subspace, then the restriction ξ|A is the family of vector spaces with total space E(ξ|A) = π −1 (A) , base space B(ξ|A) = A and projection map π −1 (A) → A given by restricting π. There is a canonical map ξ|A → ξ, given by the inclusions ˆı and i that make the diagram ˆ ı / E(ξ) E(ξ|A) π

 A

π i

 /B

commute. Definition 6.1.7. A (real) vector bundle ξ is a locally trivial family of (real) vector spaces π : E → B, meaning that B has a cover by open subsets U such that each restriction ξ|U admits a trivialization. More explicitly, this trivialization is a homeomorphism h : U ×Rn → π −1 (U ) such that x 7→ h(b, x) is a linear isomorphism from Rn to Fb (ξ), for each b ∈ U . An n-dimensional real vector bundle is also called an Rn -bundle. A 1dimensional bundle is often called a line bundle. If ξ and η are vector bundles, then a bundle map ξ → η is the same as a map of the underlying families of vector spaces. Example 6.1.8. The trivial family nB of vector spaces is a vector bundle, called the trivial vector bundle over B. A vector bundle that is isomorphic to the trivial bundle is said to be trivialized, or trivializable, or just a trivial bundle. Example 6.1.9. The tangent bundle τM of a smooth n-manifold M is an ndimensional real vector bundle π : T M → M , with fiber π −1 (p) = Tp M for p ∈ M given by the tangent space to M at that point. If τM admits a trivialization, we say that M is parallelizable. An open submanifold of Rn is parallelizable. The 2-sphere S 2 is not parallelizable.

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Example 6.1.10. Suppose that M ⊂ Rn+k is embedded as a smooth submanifold of Rn+k . The embedding identifies each tangent space Tp M with a subspace of Tp Rn+k ∼ = Rn+k . The normal bundle νM is the k-dimensional real vector bundle π : E(νM ) → M , with fiber π −1 (p) = (Tp M )⊥ consisting of the orthogonal complement of Tp M in Rn+k . The normal bundle for the usual embedding S n ⊂ Rn+1 admits a trivialization 1S n ∼ = νS n , taking (p, t) in S n × R to (p, tp) in E(νS n ), i.e., t times the outward pointing unit normal to S n at p. Example 6.1.11. Let RP n be the n-dimensional real projective space, given as the orbit space for the antipodal action on S n , or equivalently as the space of lines through the origin in Rn+1 . The canonical line bundle γn1 = γ 1 (Rn+1 ) over RP n has total space E(γn1 ) = {(L, v) | v ∈ L ⊂ Rn+1 } ⊂ RP n × Rn+1 consisting of the pairs (L, v) with L ⊂ Rn+1 a 1-dimensional subspace, and v ∈ L a point on that line. The space RP n is covered by the open subspaces Ui ∼ = Rn , consisting of the n+1 lines through the points x = (x0 , . . . , xn ) ∈ R with xi = 1, for each of the cases 0 ≤ i ≤ n. The restriction of γn1 to Ui is trivialized by the homeomorphism ∼ =

Ui × R −→ π −1 (Ui ) over Ui , taking a pair (L, t), for L ∈ Ui the line through x with xi = 1 and t ∈ R, to the pair (L, tx) in E(γn1 ). Hence γn1 is a line bundle. Similarly, there is a canonical line bundle γ 1 over RP ∞ , and the restriction of γ 1 to RP n ⊂ RP ∞ is γn1 . Definition 6.1.12. A section in a vector bundle π : E → B is a map s : B → E such that π ◦ s = idB . It takes each b ∈ B to the fiber Fb = π −1 (b) over that point. A section s is nowhere zero if the vector s(b) is not the zero vector in Fb , for each b ∈ B. A k-tuple of sections s1 , . . . , sk in π : E → B is nowhere dependent if the vectors s1 (b), . . . , sk (b) are linearly independent in Fb , for each b ∈ B. Theorem 6.1.13. The bundle γn1 over RP n is not trivial, for each n ≥ 1. Proof. A trivial line bundle admits a section that is nowhere zero. We show that every section of γn1 is zero somewhere in RP n . Let s : RP n → E(γn1 ) be a section, and consider the composite q

s

S n −→ RP n −→ E(γn1 ) that first maps x ∈ S n to the line L through x, then to the value s(L) = (L, t(x)x) of the section at that line, for some real number t(x). The function t : S n → R is continuous, and satisfies t(−x) = −t(x), since x and −x both map to s(L), so that t(−x)(−x) = t(x)x. For n ≥ 1 the space S n is connected, so by the intermediate value theorem there has to be a point x ∈ S n with t(x) = 0, which means that s(L) is the zero vector for L equal to the line though x.

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Theorem 6.1.14. An Rn -bundle ξ is trivial if and only if it admits n sections s1 , . . . , sn that are nowhere dependent. Proof. A trivialization h : B × Rn −→ E(ξ) over B determines a nowhere dependent n-tuple of sections by the formulas si (b) = h(b, ei ) for b ∈ B, where ei ∈ Rn is the i-th standard basis vector, for 1 ≤ i ≤ n. Conversely, the sections determine the trivialization by the formula h(b, x) =

n X

xi si (b)

i=1

for x = (x1 , . . . , xn ) ∈ Rn . The resulting map h is continuous, and restricts to a linear isomorphism on each fiber. It follows that h is a homeomorphism, by using that the inverse A−1 of an invertible n × n matrix A depends continuously on A. Example 6.1.15. A section in the tangent bundle T M → M of a smooth manifold is the same as a vector field on M . Hence an n-manifold M is parallelizable if and only if it admits n vector fields that are nowhere dependent. Example 6.1.16. The circle S 1 is parallelizable. One nowhere zero vector field is given by s(x) = (x, s¯(x)), where s¯(x1 , x2 ) = (−x2 , x1 ) . Note that i(x1 + ix2 ) = −x2 + ix1 in C. The 3-sphere S 3 is also parallelizable. Three nowhere dependent vector fields are given by si (x) = (x, s¯i (x)), where s¯1 (x1 , x2 , x3 , x4 ) = (−x2 , x1 , −x4 , x3 ) s¯2 (x1 , x2 , x3 , x4 ) = (−x3 , x4 , x1 , −x2 ) s¯3 (x1 , x2 , x3 , x4 ) = (−x4 , −x3 , x2 , x1 ) . Note that i(x1 +ix2 +jx3 +kx4 ) = −x2 +ix1 −jx4 +kx3 etc. in the quaternions. More generally, any Lie group G is parallelizable, since left multiplication Lg : G → G by an element g induces an isomorphism (Lg )∗ : Te G → Tg G of tangent spaces. These combine to a trivialization ∼ =

G × Te G −→ T G over G. Here Te G = g is the Lie algebra of G.

6.2

Other kinds of vector bundles

Definition 6.2.1. We define families of complex vector spaces and complex vector bundles in the same way as in the real case, by replacing real vector spaces, real linear homomorphisms and the standard n-dimensional real vector space Rn by complex vector spaces, complex linear homomorphisms and the standard n-dimension complex vector space Cn , respectively.

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Example 6.2.2. Let CP n be the n-dimensional complex projective space, given as the orbit space for the free action of the complex numbers of unit length S 1 ⊂ C on the unit sphere S 2n+1 ⊂ Cn+1 , or equivalently as the space of complex lines through the origin in Cn+1 . The canonical complex line bundle γn1 = γ 1 (Cn+1 ) over CP n has total space E(γn1 ) = {(L, v) | v ∈ L ⊂ Cn+1 } ⊂ CP n × Cn+1 consisting of the pairs (L, v) with L ⊂ Cn+1 a 1-dimensional complex subspace, and v ∈ L a point on that complex line. Similarly, there is a canonical complex line bundle γ 1 = γ 1 (C∞ ) over CP ∞ , and the restriction of γ 1 to CP n ⊂ CP ∞ is γn1 . Definition 6.2.3. A real inner product space is a real vector space V with a symmetric, bilinear and positive definite pairing V × V → R, taking (v, w) to hv, wi. A complex inner product space is a complex vector space V with a pairing V × V → C, taking (v, w) to hv, wi, that is conjugate symmetric, complex linear in the first variable, and positive definite. A linear homomorphism f : V → W of (real or complex) inner product spaces is an isometry if hf (v), f (v 0 )i = hv, v 0 i for all v, v 0 ∈ V . Such a homomorphism is necessarily injective. Example 6.2.4. The Euclidean real inner product on Rn takes (x1 , . . . , xn ) and (y1 , . . . , yn ) to n X xi yi . i=1

The Hermitian complex inner product on Cn takes (x1 , . . . , xn ) and (y1 , . . . , yn ) to n X xi y¯i . i=1

Definition 6.2.5. Euclidean vector bundles are defined like real vector bundles, replacing real vector spaces by real inner products spaces, real linear homomorphisms by isometries, and equipping the standard vector space Rn with the Euclidean inner product. Hermitian vector bundles are defined like complex vector bundles, replacing complex vector spaces by complex inner products spaces, complex linear homomorphisms by isometries, and equipping the standard vector space Cn with the Hermitian inner product. Example 6.2.6. Suppose again that M ⊂ Rn+k is embedded as a smooth n-dimensional submanifold. The Euclidean inner product on Tp Rn+k ∼ = Rn+k n+k restricts to an Euclidean inner product on the subspace Tp M ⊂ Tp R , for each p ∈ M . Hence τM becomes a Euclidean vector bundle, and the bundle map τM → τRn+k is an isometry.

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6.3

77

Constructing new bundles out of old

Definition 6.3.1. Let π : E → B be a vector bundle. If f : A → B is any map, then the pullback f ∗ ξ is the vector bundle π : E(f ∗ ξ) → A, where E(f ∗ ξ) = {(a, e) ∈ A × E | f (a) = π(e)} is the fiber product of A and E = E(ξ) over B = B(ξ), and π(a, e) = a. We give Fa (f ∗ ξ) = π −1 (a) the vector space structure that makes the homeomorphism Fa (f ∗ ξ) ∼ = Ff (a) (ξ) given by (a, e) 7→ e a linear isomorphism. To prove that f ∗ ξ is a vector bundle, one can check that if the restriction of ξ to U ⊂ B admits a trivialization, then the restriction of f ∗ ξ to f −1 (U ) ⊂ A also admits a trivialization. There is a canonical map f ∗ ξ → ξ, given by the maps fˆ: E(f ∗ ξ) → E(ξ) taking (a, e) to e and f : A → B, making the diagram fˆ

E(f ∗ ξ)

/ E(ξ) π

π

 A

f

 /B

commute. If η → ξ is any bundle map (g, g¯), with g¯ = f : A → B, then it factors uniquely as the composite of of a bundle map (k, idA ) : η → f ∗ ξ over A and the canonical map (fˆ, f ) : f ∗ ξ → ξ: g

E(η)

k

fˆ

 / E(ξ)

f

 /B H

π

π

 A

/ E(f ∗ ξ)

idA

π

 /A g ¯

Here k(e) = (π(e), g(e)) ∈ E(f ∗ ξ) ⊂ A × E(ξ) for e ∈ E(η). Example 6.3.2. If i : A → B is an inclusion, then the pullback i∗ ξ is naturally isomorphic to the restriction ξ|A. Example 6.3.3. If f : A → B is a constant map to a point b ∈ B, then f ∗ ξ is isomorphic to the trivial bundle nA , since E(f ∗ ξ) = A × Fb (ξ) and π : E(f ∗ ξ) → B(f ∗ ξ) = A takes (a, e) to a. Example 6.3.4. The pullback of the canonical line bundle γn1 over RP n along the covering map q : S n → RP n is the normal bundle of S n ⊂ Rn+1 , which is trivial: q ∗ γn1 ∼ = νS n ∼ = 1S n . Example 6.3.5. The pullbacks of complex, Euclidean or Hermitian vector bundles are similarly defined.

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Definition 6.3.6. The Cartesian product ξ1 × ξ2 of two vector bundles ξ1 and ξ2 , with projection maps π1 : E(ξ1 ) → B(ξ1 ) and π2 : E(ξ2 ) → B(ξ2 ), is the bundle with projection map π1 × π2 : E(ξ1 ) × E(ξ2 ) −→ B(ξ1 ) × B(ξ2 ) . The fiber F(b1 ,b2 ) (ξ1 × ξ2 ) = Fb1 (ξ1 ) × Fb2 (ξ2 ) has the vector space structure given by the product on the right. Definition 6.3.7. Let ξ and η be vector bundles over the same base space B. The Whitney sum ξ ⊕ η is the vector bundle over B defined by the pullback ξ ⊕ η = ∆∗ (ξ × η) of the Cartesian product ξ × η along the diagonal map ∆ : B → B × B. The fiber Fb (ξ ⊕ η) = Fb (ξ) × Fb (η) is isomorphic to the direct sum Fb (ξ) ⊕ Fb (η), for each b ∈ B. n ∼ Example 6.3.8. There are natural isomorphisms ξ ⊕ η ∼ = η ⊕ ξ and m B ⊕ B = m+n . B

Example 6.3.9. We can recover the Cartesian product from the Whitney sum, as there is a canonical isomorphism ξ1 × ξ2 ∼ = p∗1 (ξ1 ) ⊕ p∗2 (ξ2 ) where pi : B(ξ1 ) × B(ξ2 ) → B(ξi ) is the projection to the i-th factor, for i = 1, 2. Example 6.3.10. The Cartesian product and Whitney sum of complex, Euclidean and Hermitian vector bundles are similarly defined. If V and W are inner product spaces, then the inner product on V ⊕ W ∼ = V × W is given by h(v, w), (v 0 , w0 )i = hv, v 0 i + hw, w0 i . This is also known as the orthogonal sum V ⊥ W . Definition 6.3.11. Let V be the topological category of finite dimensional vector spaces and linear isomorphisms. This means that for each pair of vector spaces V , W the set V (V, W ) of linear isomorphisms V ∼ = W is topologized as an open subspace of the vector space of linear homomorphisms V → W . The composition rule V (V, W ) × V (U, V ) → V (U, W ) is then continuous. Let T : V × · · · × V → V be a continuous functor in k variables. This means that the rule V (V1 , W1 ) × · · · × V (Vk , Wk ) −→ V (T (V1 , . . . , Vk ), T (W1 , . . . , Wk )) that assigns the induced isomorphism T (ϕ1 , . . . , ϕk ) : T (V1 , . . . , Vk ) ∼ = T (W1 , . . . , Wk ) to each k-tuple of isomorphisms ϕ1 : V1 ∼ = W 1 , . . . , ϕk : V k ∼ = Wk , is a continuous map.

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Let ξ1 , . . . , ξk be a k-tuple of vector bundles over the same base space B. We define the vector bundle T (ξ1 , . . . , ξk ) over B to have fibers Fb = T (Fb (ξ1 ), . . . , Fb (ξk )) , `

total space E = b∈B Fb , and projection π : E → B taking all of Fb to b. The topology on E is determined as follows. Each point b ∈ B has an open neighborhood U such that each bundle ξi |U admits a trivialization hi : nUi ∼ = ξi |U , for some non-negative integers n1 , . . . , nk ≥ 0. The linear isomorphisms ϕi = hi,b : Rni ∼ = Fb (ξi ) for b ∈ U induce a linear isomorphism T (ϕ1 , . . . , ϕk ) : T (Rn1 , . . . , Rnk ) ∼ = Fb for each b ∈ U , and these combine to a bijective function h : U × T (Rn1 , . . . , Rnk ) −→ π −1 (U ) over U . We give π −1 (U ) the topology that makes h a homeomorphism. Since T is a continuous functor, this topology does not depend on the choices of trivializations hi . These subsets π −1 (U ) cover E, and we give E the finest topology that makes each inclusion π −1 (U ) → E continuous. Definition 6.3.12. Let η be a Euclidean vector bundle, and let ξ ⊂ η be a subbundle over the same base space B. Let Fb (ξ ⊥ ) ⊂ Fb (η) be the orthogonal complement of Fb (ξ), for each b ∈ B, and let E(ξ ⊥ ) ⊂ E(η) be the disjoint union of the fibers Fb (ξ ⊥ ). Then ξ ⊥ ⊂ η is a subbundle, and the canonical map ∼ =

ξ ⊕ ξ ⊥ −→ η given by the sum Fb (ξ) ⊕ Fb (ξ ⊥ ) → Fb (η) in each fiber, is an isomorphism. We call ξ ⊥ the orthogonal complement of ξ in η. Example 6.3.13. Let ξ be a real vector bundle, and let Fb (ξ ∗ ) = HomR (Fb (ξ), R) be the linear dual to Fb (ξ), for each b ∈ B. Let E(ξ ∗ ) be the disjoint union of the fibers Fb (ξ ∗ ). Then ξ ∗ is a vector bundle. If ξ is a Euclidean bundle, then the inner product on Fb (ξ) induces a linear isomorphism Fb (ξ) → HomR (Fb (ξ), R) in each fiber. These combine to an isomorphism ∼ = ξ −→ ξ ∗ of vector bundles. Example 6.3.14. Let ξ be a complex vector bundle, and let Fb (ξ ∗ ) = HomC (Fb (ξ), C) be the linear dual to Fb (ξ), for each b ∈ B. Let E(ξ ∗ ) be the disjoint union of the fibers Fb (ξ ∗ ). Then ξ ∗ is a complex vector bundle. Also let ¯ = Fb (ξ) Fb (ξ)

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be the complex conjugate of Fb (ξ), meaning that multipliction by z = x+iy ∈ C acts like multiplication by z¯ = x−iy in Fb (ξ). Then ξ¯ is a complex vector bundle. If ξ is a Hermitian bundle, then the inner product on Fb (ξ) induces a linear isomorphism Fb (ξ) → HomC (Fb (ξ), C) in each fiber, taking w to the homomorphism v 7→ hv, wi. These combine to an isomorphism ∼ = ξ¯ −→ ξ ∗

of complex vector bundles. ((It is not generally the case that ξ ∼ = ξ ∗ for complex ξ.))

6.4

Grassmann manifolds and universal bundles

Definition 6.4.1. An n-tuple of linearly independent vectors x1 , . . . , xn in Rn+k called an n-frame. The set of n-frames in Rn+k is an open subspace Vn (Rn+k ) ⊂ Rn+k × · · · × Rn+k of Euclidean n(n + k)-space, called a Stiefel manifold. Let the Grassmann manifold Gn (Rn+k ) be the set of all n-dimensional vector subspaces X of Rn+k . There is a canonical map q : Vn (Rn+k ) −→ Gn (Rn+k ) that takes an n-frame x1 , . . . , xn to the n-dimensional subspace X that it spans. We give Gn (Rn+k ) the quotient topology determined by q, so a subset U ⊂ Gn (Rn+k ) is open if and only if the preimage q −1 (U ) is open in Vn (Rn+k ). Lemma 6.4.2. Gn (Rn+k ) is a closed nk-dimensional manifold. Proof. Let X ∈ Gn (Rn+k ) be any n-dimensional subspace of Rn+k , with kdimensional orthogonal complement X ⊥ . We can identify X × X ⊥ with X ⊕ X⊥ ∼ = Rn+k . The graph of any linear homomorphism f : X → X ⊥ is an ndimensional subspace of X × X ⊥ ∼ = Rn+k . The n-dimensional subspaces of n+k R that arise in this way form an open neighborhood of X in Gn (Rn+k ), and this space is homeomorphic to the space of linear homomorphisms Rn → Rk , which is homeomorphic to Rnk . The subspace VnO (Rn+k ) ⊂ Vn (Rn+k ) of orthonormal n-tuples is a closed subspace VnO (Rn+k ) ⊂ S n+k−1 × · · · × S n+k−1 of the compact space on the right, hence is compact. The restriction q O : VnO (Rn+k ) → Gn (Rn+k ) defines the same topology as q, hence Gn (Rn+k ) is compact. Example 6.4.3. G1 (Rn+1 ) is the same as RP n .

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Definition 6.4.4. The canonical Rn -bundle γ n (Rn+k ) over Gn (Rn+k ) has total space E(γ n (Rn+k )) = {(X, v) | v ∈ X ⊂ Rn+k } ⊂ Gn (Rn+k ) × Rn+k consisting of the pairs (X, v) with X ⊂ Rn+k an n-dimensional subspace, and v ∈ X a vector in that subspace. The projection map π : E(γ n (Rn+k )) → Gn (Rn+k ) is given by π(X, v) = X, and the fiber FX = π −1 (X) is identified with the vector space X itself, via the linear isomorphism v 7→ (X, v). Lemma 6.4.5. γ n (Rn+k ) is a vector bundle. ((ETC))

6.5

Oriented bundles and the Euler class

6.6

The Thom isomorphism theorem

6.7

Chern classes

6.8

Pontryagin classes

((ETC))

6.9 6.9.1

Bordism Thom complexes and the Thom isomorphism S(ξ) '

 E0 (ξ)

/ D(ξ) '

 / E(ξ)  B

/ (T h(ξ), ∗)  / (E(ξ), E0 (ξ))

CHAPTER 6. VECTOR BUNDLES AND CLASSIFYING SPACES

6.9.2

82

Tubular neighborhoods and the Pontryagin–Thom construction

M smooth, oriented n-manifold.

TM

S n+k o

S(ν) /

S n+k −→

6.9.3

E(˜ γ n (Rn+k )) 8 q qqq q q q qqq

 ˜ n (Rn+k ) G O r8 τM rrr r rrr  rrr o M L ∼ = O LL LLL LL νM LLL &  ˜ k (Rn+k ) G O

o Rn+k o O

O / D(ν) /

/ NM LLL LLL LLL LL& E(˜ γ k (Rn+k ))

/ E(˜ γn)

 /G ˜n

/G ˜k O

/ E(˜ γk)

S n+k D(νM ) ∼ = T h(νM ) −→ T h(˜ γ k ) = M SOk = S n+k − Int D(νM ) S(νM )

Transversality and Thom’s theorem

˜ k → D(˜ s0 : BSO(k) = G γk ) → T h(˜ γk ) = M SOk has normal bundle γ˜k . Any base-point preserving map S n+k → T h(˜ γk ) is homotopic to one that ˜ k → T h(˜ ˜ k ) is then an is transverse to s0 : G γk ). The preimage M = g −1 (G n-dimensional oriented submanifold of Rn+k ⊂ S n+k . M 

f

γ ˜k

νM



S n+k

/G ˜k 

g

 / T h(˜ γk)

If H : S n+k × I → T h(˜ γk ) is a base-point preserving homotopy from g to ˜ k , then the preimage W = H −1 (G ˜ k ) is a g , all of which are transverse to G 0 0 −1 ˜ cobordism from M to M = (g ) (Gk ). Oriented bordism: 0

∼ ΩSO γ k ) = πn (M SO) . n = colim πn+k T h(˜ k

CHAPTER 6. VECTOR BUNDLES AND CLASSIFYING SPACES

6.9.4

83

Homology of Thom spectra

˜ n+k (S n+k ) H

∼ =

πn+k T h(νM )

/ πn+k T h(˜ γk)

/ πn M SO

 /H ˜ n+k (T h(νM ))

 /H ˜ n+k (T h(˜ γ k ))

 / Hn (M SO)

(−)∩uνM

∼ =

 Hn (M )

(−)∩uγ˜ k

∼ =

 / Hn (G ˜k)

f∗

∼ =

 / Hn (BSO)

The Hurewicz image in Hn (M SO) of homotopy class in πn (M SO) ∼ = ΩSO n is ˜ n+k (S n+k ), which under the Thom the image of fundamental class [S n+k ] ∈ H isomorphism Hn (M SO) ∼ = Hn (BSO) corresponds to the image of the funda˜ k → BSO for mental class [M ] ∈ Hn (M ) under the classifying map f : M → G the normal bundle νM . Serre: ∼ = πn (M SO) ⊗ Q −→ Hn (M SO) ⊗ Q

6.9.5

Pontryagin numbers

Consider (co-)homology with coefficients in a ring R containing Z[1/2]. Recall that ˜ 2m ; R) ∼ H ∗ (G = R[p1 , . . . , pm−1 , e] with pm = e2 , and

˜ 2m+1 ; R) ∼ H ∗ (G = R[p1 , . . . , pm ]

with e = 0. The limit systems ∼ = ˜ k ; R) −→ colim Hn (G Hn (BSO; R) k

and

∼ =

˜ k ; R) H n (BSO; R) −→ lim H n (G k

˜ k ; R) ∼ ˜ k ; R), R), stabilize for a finite k, so the isomorphism Hn (G = Hom(H n (G adjoint to the evaluation pairing, stabilizes to an isomorphism Hn (BSO; R) ∼ = Hom(H n (BSO; R), R) . Here

H ∗ (BSO; R) ∼ = R[pi | i ≥ 1] .

Hence H n (BSO; R) = 0 unless n is divisible by four. When n = 4k, it is the free R-module generated by the monomials pI = pi11 · . . . · pikk where I = (i1 , . . . , ik ) is a k-tuple of non-negative integers such that i1 + 2i2 + · · · + kik = k .

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These correspond to partitions of k, i.e., unordered ways of writing k as a sum of natural numbers, with ij counting how often j occurs in the sum. An element α ∈ Hn (BSO) is thus in one-to-one correspondence with the collection of numbers hpI , αi ∈ R where I ranges over these k-tuples indexing partitions of k. Let f : M → BSO be the classifying map for the stable normal bundle of M , so that pi (νM ) = f ∗ (pi ) for each i. When α = f∗ [M ], we can rewrite these numbers as hpI , f∗ [M ]i = hf ∗ (pI ), [M ]i = hpI (νM ), [M ]i where pI (νM ) = pi11 (νM ) · . . . · pikk (νM ) is a monomial in the Pontryagin classes of the normal bundle of M . The numbers hpI (νM ), [M ]i ∈ R are called the normal Pontryagin numbers of M . Theorem 6.9.1. ΩSO n ⊗ Q = 0 for n not divisible by four. When n = 4k ≥ 0, the rule taking the oriented cobordism class of M to the collection of normal Pontryagin numbers hpI (νM ), [M ]i ∈ Q P for I = (i1 , . . . , ik ) with j jij = k, defines an isomorphism M ∼ = ΩSO Q. n ⊗ Q −→ I

In view of the formula τM ⊕ νM ∼ = n+k and its consequence p(τM ) ∪ p(νM ) = 1 ∗

in H (M ; R), it follows that each monomial pI (νM ) can be written as a polynomial in the tangential Pontryagin classes pi (τM ), and vice versa. We can therefore use the monomials pI (τM ) = pi11 (τM ) · . . . · pikk (τM ) in the tangential Pontryagin classes of M as alternative operators to detect Hn (M ; R), in place of the normal classes. Theorem 6.9.2. ΩSO n ⊗ Q = 0 for n not divisible by four. When n = 4k ≥ 0, the rule taking the oriented cobordism class of M to the collection of (tangential) Pontryagin numbers pI (M ) = hpI (τM ), [M ]i ∈ Q P for I = (i1 , . . . , ik ) with j jij = k, defines an isomorphism M ∼ = ΩSO Q. n ⊗ Q −→ I

6.9.6

Computations in low dimensions

6.9.7

The index formula

Bibliography [1] Pierre E. Conner, Differentiable periodic maps, 2nd ed., Lecture Notes in Mathematics, vol. 738, Springer, Berlin, 1979. [2] Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. [3] John W. Milnor and James D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. Annals of Mathematics Studies, No. 76.

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