Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Tiling Spaces and Topology Lorenzo Sadun

Trondheim, June 8, 2015

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Outline 1

Tilings

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Outline 1

Tilings

2

Types of tilings Hierarchical tilings Cut and project tilings Local matching rules

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Outline 1

Tilings

2

Types of tilings Hierarchical tilings Cut and project tilings Local matching rules

3

Local topology of tiling spaces

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Outline 1

Tilings

2

Types of tilings Hierarchical tilings Cut and project tilings Local matching rules

3

Local topology of tiling spaces

4

Inverse limits

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Outline 1

Tilings

2

Types of tilings Hierarchical tilings Cut and project tilings Local matching rules

3

Local topology of tiling spaces

4

Inverse limits

5

Cohomology

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Table of Contents 1

Tilings

2

Types of tilings Hierarchical tilings Cut and project tilings Local matching rules

3

Local topology of tiling spaces

4

Inverse limits

5

Cohomology

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Chair tiling

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Penrose

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

A bathroom floor

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Rules of the Game

Finitely many tile types

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Rules of the Game

Finitely many tile types Each type described by a geometric set (prototile) and label.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Rules of the Game

Finitely many tile types Each type described by a geometric set (prototile) and label. Tiles are translates of prototiles. (Translate the set, keep the label)

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Rules of the Game

Finitely many tile types Each type described by a geometric set (prototile) and label. Tiles are translates of prototiles. (Translate the set, keep the label) Finite local complexity (FLC) with respect to translation.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Rules of the Game

Finitely many tile types Each type described by a geometric set (prototile) and label. Tiles are translates of prototiles. (Translate the set, keep the label) Finite local complexity (FLC) with respect to translation. I.e., finitely many tile types, finitely many ways for 2 tiles to meet.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Rules of the Game

Finitely many tile types Each type described by a geometric set (prototile) and label. Tiles are translates of prototiles. (Translate the set, keep the label) Finite local complexity (FLC) with respect to translation. I.e., finitely many tile types, finitely many ways for 2 tiles to meet. WLOG, can assume tiles are polyhedra meeting full face to full face.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Hierarchical tilings Cut and project tilings Local matching rules

Table of Contents 1

Tilings

2

Types of tilings Hierarchical tilings Cut and project tilings Local matching rules

3

Local topology of tiling spaces

4

Inverse limits

5

Cohomology

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Hierarchical tilings Cut and project tilings Local matching rules

Substitutions 1-dimensional example (Fibonacci) : a → ab, b → a.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Hierarchical tilings Cut and project tilings Local matching rules

Substitutions 1-dimensional example (Fibonacci) : a → ab, b → a. a

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Hierarchical tilings Cut and project tilings Local matching rules

Substitutions 1-dimensional example (Fibonacci) : a → ab, b → a. a ab

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Hierarchical tilings Cut and project tilings Local matching rules

Substitutions 1-dimensional example (Fibonacci) : a → ab, b → a. a ab ab.a

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Hierarchical tilings Cut and project tilings Local matching rules

Substitutions 1-dimensional example (Fibonacci) : a → ab, b → a. a ab ab.a ab.a.ab

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Hierarchical tilings Cut and project tilings Local matching rules

Substitutions 1-dimensional example (Fibonacci) : a → ab, b → a. a ab ab.a ab.a.ab ab.a.ab.ab.a

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Hierarchical tilings Cut and project tilings Local matching rules

Substitutions 1-dimensional example (Fibonacci) : a → ab, b → a. a ab ab.a ab.a.ab ab.a.ab.ab.a ab.a.ab.ab.a.ab.a.ab

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Hierarchical tilings Cut and project tilings Local matching rules

Substitutions 1-dimensional example (Fibonacci) : a → ab, b → a. a ab ab.a ab.a.ab ab.a.ab.ab.a ab.a.ab.ab.a.ab.a.ab A word is legal if it sits inside one of these patterns.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Hierarchical tilings Cut and project tilings Local matching rules

Substitutions 1-dimensional example (Fibonacci) : a → ab, b → a. a ab ab.a ab.a.ab ab.a.ab.ab.a ab.a.ab.ab.a.ab.a.ab A word is legal if it sits inside one of these patterns. A bi-infinite word is legal if every sub-word is legal.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Hierarchical tilings Cut and project tilings Local matching rules

Substitutions 1-dimensional example (Fibonacci) : a → ab, b → a. a ab ab.a ab.a.ab ab.a.ab.ab.a ab.a.ab.ab.a.ab.a.ab A word is legal if it sits inside one of these patterns. A bi-infinite word is legal if every sub-word is legal. √ Make into self-similar tilings by assigning length (1 + 5)/2 to a tile and 1 to b tile.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Hierarchical tilings Cut and project tilings Local matching rules

Chair tiling

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Hierarchical tilings Cut and project tilings Local matching rules

Fusions

Much like substitutions, only Build from the ground up. Rules for building (n + 1)-supertiles out of n-supertiles. Rules can be different at different levels. Much (but not all) of the theory of substitution tilings goes through.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Hierarchical tilings Cut and project tilings Local matching rules

Fibonacci as fusion

Two tiles, a and b. P0 (a) = a, P0 (b) = b.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Hierarchical tilings Cut and project tilings Local matching rules

Fibonacci as fusion

Two tiles, a and b. P0 (a) = a, P0 (b) = b. Fusion rule: Pn+1 (a) = Pn (a)Pn (b); Pn+1 (b) = Pn (a).

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Hierarchical tilings Cut and project tilings Local matching rules

Fibonacci as fusion

Two tiles, a and b. P0 (a) = a, P0 (b) = b. Fusion rule: Pn+1 (a) = Pn (a)Pn (b); Pn+1 (b) = Pn (a). P1 (a) = ab; P1 (b) = a.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Hierarchical tilings Cut and project tilings Local matching rules

Fibonacci as fusion

Two tiles, a and b. P0 (a) = a, P0 (b) = b. Fusion rule: Pn+1 (a) = Pn (a)Pn (b); Pn+1 (b) = Pn (a). P1 (a) = ab; P1 (b) = a. P2 (a) = aba; P2 (b) = ab.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Hierarchical tilings Cut and project tilings Local matching rules

Fibonacci as fusion

Two tiles, a and b. P0 (a) = a, P0 (b) = b. Fusion rule: Pn+1 (a) = Pn (a)Pn (b); Pn+1 (b) = Pn (a). P1 (a) = ab; P1 (b) = a. P2 (a) = aba; P2 (b) = ab. P3 (a) = abaab; P3 (b) = aba.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Hierarchical tilings Cut and project tilings Local matching rules

Fibonacci as fusion

Two tiles, a and b. P0 (a) = a, P0 (b) = b. Fusion rule: Pn+1 (a) = Pn (a)Pn (b); Pn+1 (b) = Pn (a). P1 (a) = ab; P1 (b) = a. P2 (a) = aba; P2 (b) = ab. P3 (a) = abaab; P3 (b) = aba. P3 (a) = abaababa; P3 (b) = abaab.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Hierarchical tilings Cut and project tilings Local matching rules

The setup Embed Rd in higher-dimensional space in an irrational way. Start with period array in Rd+k . Restrict to strip defined by window in Rk . Project points onto Rd . Convert to tiling if desired.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Hierarchical tilings Cut and project tilings Local matching rules

The setup Embed Rd in higher-dimensional space in an irrational way. Start with period array in Rd+k . Restrict to strip defined by window in Rk . Project points onto Rd . Convert to tiling if desired.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Hierarchical tilings Cut and project tilings Local matching rules

Properties

Pure point diffraction. Measurably conjugate to torus rotation. Close connections to number theory. Topology of tiling space ⇔ Geometry of window.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Hierarchical tilings Cut and project tilings Local matching rules

Wang tiles

Square tiles with colored edges. Two tiles can touch iff edges have same color. Berger ‘66: Can have horribly complicated set that tiles plane but not periodically. Smaller set by Rafael Robinson that you can actually understand.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Hierarchical tilings Cut and project tilings Local matching rules

Jigsaw puzzles

Not just squares. Use whatever shapes you want. Implement matching rules with notches and prongs. If it fits, it’s legal. Most famous example: Penrose.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Hierarchical tilings Cut and project tilings Local matching rules

Substitutions and matching rules

Theorem (Robinson, Mozes, Radin, Goodman-Strauss) Every substitution tiling space (within reason) is an almost 1:1 factor of a local matching rules space.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Hierarchical tilings Cut and project tilings Local matching rules

Substitutions and matching rules

Theorem (Robinson, Mozes, Radin, Goodman-Strauss) Every substitution tiling space (within reason) is an almost 1:1 factor of a local matching rules space. Unfortunately, almost nothing is known about the topology of local matching rules spaces.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Hierarchical tilings Cut and project tilings Local matching rules

Substitutions vs. cut-and-project

Substitution tilings are cut-and-project if and only if they have pure point spectrum. (Most don’t)

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Hierarchical tilings Cut and project tilings Local matching rules

Substitutions vs. cut-and-project

Substitution tilings are cut-and-project if and only if they have pure point spectrum. (Most don’t) Most cut-and-project tilings are not substitutions.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Hierarchical tilings Cut and project tilings Local matching rules

Substitutions vs. cut-and-project

Substitution tilings are cut-and-project if and only if they have pure point spectrum. (Most don’t) Most cut-and-project tilings are not substitutions. Many famous examples are both.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Hierarchical tilings Cut and project tilings Local matching rules

Substitutions vs. cut-and-project

Substitution tilings are cut-and-project if and only if they have pure point spectrum. (Most don’t) Most cut-and-project tilings are not substitutions. Many famous examples are both. Penrose is substitution and cut-and-project and local matching rules.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Table of Contents 1

Tilings

2

Types of tilings Hierarchical tilings Cut and project tilings Local matching rules

3

Local topology of tiling spaces

4

Inverse limits

5

Cohomology

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Tiling metric

Two tilings are -close if they agree on B1/ up to rigid translations by < .

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Tiling metric

Two tilings are -close if they agree on B1/ up to rigid translations by < . In other settings, allow small rigid motions instead of translations, or small motions of each tile independent of the rest. Just not in this talk!

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Neighborhood of a tiling

To get a tiling in an -neighborhood of T :

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Neighborhood of a tiling

To get a tiling in an -neighborhood of T : Erase all of T outside of B1/ .

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Neighborhood of a tiling

To get a tiling in an -neighborhood of T : Erase all of T outside of B1/ . Translate by up to .

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Neighborhood of a tiling

To get a tiling in an -neighborhood of T : Erase all of T outside of B1/ . Translate by up to . Fill in the tiles near ∞. (Many discrete choices)

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Neighborhood of a tiling

To get a tiling in an -neighborhood of T : Erase all of T outside of B1/ . Translate by up to . Fill in the tiles near ∞. (Many discrete choices) Neigborhood looks like Rd × C .

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Tiling Spaces

A Tiling Space is a set Ω of tilings such that:

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Tiling Spaces

A Tiling Space is a set Ω of tilings such that: Ω is translation-invariant.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Tiling Spaces

A Tiling Space is a set Ω of tilings such that: Ω is translation-invariant. Ω is complete w.r.t. tiling metric.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Tiling Spaces

A Tiling Space is a set Ω of tilings such that: Ω is translation-invariant. Ω is complete w.r.t. tiling metric. (Sometimes require non-periodicity: T − x = T ⇒ x = 0)

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Tiling Spaces

A Tiling Space is a set Ω of tilings such that: Ω is translation-invariant. Ω is complete w.r.t. tiling metric. (Sometimes require non-periodicity: T − x = T ⇒ x = 0) Canonical example: ΩT = {T − x|x ∈ Rd } T 0 ∈ ΩT iff every finite patch of T 0 appears somewhere in T .

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Repetitivity and Minimality

A tiling T is repetitive if for each finite patch P there is a radius R(P) such that every ball of radius R contains a copy of P.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Repetitivity and Minimality

A tiling T is repetitive if for each finite patch P there is a radius R(P) such that every ball of radius R contains a copy of P. A dynamical system Ω is minimal if every orbit is dense. Theorem ΩT is minimal if and only if T is repetitive.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Table of Contents 1

Tilings

2

Types of tilings Hierarchical tilings Cut and project tilings Local matching rules

3

Local topology of tiling spaces

4

Inverse limits

5

Cohomology

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Inverse limits in general

If X0 , X1 , . . . are spaces and ρn : Xn → Xn−1 are continuous maps, Y X = lim Xi := {(x0 , x1 , . . .) ∈ Xn |ρn (xn ) = xn−1 ∀n}. ← −

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Inverse limits in general

If X0 , X1 , . . . are spaces and ρn : Xn → Xn−1 are continuous maps, Y X = lim Xi := {(x0 , x1 , . . .) ∈ Xn |ρn (xn ) = xn−1 ∀n}. ← − Xn is called n-th approximant to X , since xn determines (x0 , . . . , xn ).

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Dyadic Solenoid Example of inverse limit space. Take Xn = R/(2n Z) ' S 1 .

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Dyadic Solenoid Example of inverse limit space. Take Xn = R/(2n Z) ' S 1 . ρn induced by identity on R. Winds Xn twice around Xn−1 .

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Dyadic Solenoid Example of inverse limit space. Take Xn = R/(2n Z) ' S 1 . ρn induced by identity on R. Winds Xn twice around Xn−1 .

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Tiling spaces are inverse limits

CW complex Γn describes tiling out to distance that grows with n.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Tiling spaces are inverse limits

CW complex Γn describes tiling out to distance that grows with n. ρn is forgetful map.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Tiling spaces are inverse limits

CW complex Γn describes tiling out to distance that grows with n. ρn is forgetful map. Many different schemes: different details, same strategy.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Tiling spaces are inverse limits

CW complex Γn describes tiling out to distance that grows with n. ρn is forgetful map. Many different schemes: different details, same strategy. lim Γn = consistent instructions for tiling bigger and bigger ← − regions, i.e. instructions for a complete tiling.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Tiling spaces are inverse limits

CW complex Γn describes tiling out to distance that grows with n. ρn is forgetful map. Many different schemes: different details, same strategy. lim Γn = consistent instructions for tiling bigger and bigger ← − regions, i.e. instructions for a complete tiling. So how do instructions for partial tilings turn into a CW complex?!

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Anderson-Putnam Complex

To place a tile at the origin, need:

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Anderson-Putnam Complex

To place a tile at the origin, need: Choice of tile type ti .

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Anderson-Putnam Complex

To place a tile at the origin, need: Choice of tile type ti . Choice of point in ti to associate with origin.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Anderson-Putnam Complex

To place a tile at the origin, need: Choice of tile type ti . Choice of point in ti to associate with origin. What if origin is on boundary of 2 (or more tiles)? Identify!

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Anderson-Putnam Complex

To place a tile at the origin, need: Choice of tile type ti . Choice of point in ti to associate with origin. What if origin is on boundary of 2 (or more tiles)? Identify! a Γ0 = ti / ∼ is the Anderson-Putnam complex.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Collared tiles

Start with a tiling T . Equivalent tiles have same label and same pattern of immediate neighbors.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Collared tiles

Start with a tiling T . Equivalent tiles have same label and same pattern of immediate neighbors. Equivalence classes are called collared tiles.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Collared tiles

Start with a tiling T . Equivalent tiles have same label and same pattern of immediate neighbors. Equivalence classes are called collared tiles. Relabeling tiling with collared tiles is local operation. Does not change space.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Collared tiles

Start with a tiling T . Equivalent tiles have same label and same pattern of immediate neighbors. Equivalence classes are called collared tiles. Relabeling tiling with collared tiles is local operation. Does not change space. Can be repeated to get n-times collared tiles.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Collared Fibonacci Fibonacci sequence in 1D contains . . . abaababaabaababaababa . . .

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Collared Fibonacci Fibonacci sequence in 1D contains . . . abaababaabaababaababa . . .

Only one “b” collared tile: B = (a)b(a).

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Collared Fibonacci Fibonacci sequence in 1D contains . . . abaababaabaababaababa . . .

Only one “b” collared tile: B = (a)b(a). Three “a” collared tiles: A1 = (b)a(b), A2 = (a)a(b), A3 = (b)a(a).

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Collared Fibonacci Fibonacci sequence in 1D contains . . . abaababaabaababaababa . . .

Only one “b” collared tile: B = (a)b(a). Three “a” collared tiles: A1 = (b)a(b), A2 = (a)a(b), A3 = (b)a(a). Sequence becomes . . . BA3 A2 BA1 BA3 A2 BA3 A2 BA1 BA3 A2 BA1 B . . .

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Collared Fibonacci Fibonacci sequence in 1D contains . . . abaababaabaababaababa . . .

Only one “b” collared tile: B = (a)b(a). Three “a” collared tiles: A1 = (b)a(b), A2 = (a)a(b), A3 = (b)a(a). Sequence becomes . . . BA3 A2 BA1 BA3 A2 BA3 A2 BA1 BA3 A2 BA1 B . . .

Collared tiles have same size as regular tiles, but carry more info. Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

G¨ahler’s construction

Let Γn be the Anderson-Putnam complex of n-collared tiles.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

G¨ahler’s construction

Let Γn be the Anderson-Putnam complex of n-collared tiles. Point in Γn describes tile at origin plus nth nearest neighbors.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

G¨ahler’s construction

Let Γn be the Anderson-Putnam complex of n-collared tiles. Point in Γn describes tile at origin plus nth nearest neighbors. Edge identification can reduce that to n − 1. No sweat.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

G¨ahler’s construction

Let Γn be the Anderson-Putnam complex of n-collared tiles. Point in Γn describes tile at origin plus nth nearest neighbors. Edge identification can reduce that to n − 1. No sweat. Ω = lim Γn . ← −

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

G¨ahler’s construction

Let Γn be the Anderson-Putnam complex of n-collared tiles. Point in Γn describes tile at origin plus nth nearest neighbors. Edge identification can reduce that to n − 1. No sweat. Ω = lim Γn . ← − Conceptually very powerful idea. Great for proving theorems.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

G¨ahler’s construction

Let Γn be the Anderson-Putnam complex of n-collared tiles. Point in Γn describes tile at origin plus nth nearest neighbors. Edge identification can reduce that to n − 1. No sweat. Ω = lim Γn . ← − Conceptually very powerful idea. Great for proving theorems. Calculationally not so much, since Γn ’s are all different.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Anderson-Putnam

Applies to substitutions that “force the border”.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Anderson-Putnam

Applies to substitutions that “force the border”. Let Γn be Anderson-Putnam complex of n-supertiles.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Anderson-Putnam

Applies to substitutions that “force the border”. Let Γn be Anderson-Putnam complex of n-supertiles. All Γn ’s are the same, up to scale. Ω = lim(Γ, σ). One approximant. One expansive map. ← −

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Anderson-Putnam

Applies to substitutions that “force the border”. Let Γn be Anderson-Putnam complex of n-supertiles. All Γn ’s are the same, up to scale. Ω = lim(Γ, σ). One approximant. One expansive map. ← − To get border forcing, collar once (if necessary).

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Other techniques

Various tricks to collar as little as possible. Barge-Diamond-Hunton-Sadun. Don’t collar tiles. Collar points.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Other techniques

Various tricks to collar as little as possible. Barge-Diamond-Hunton-Sadun. Don’t collar tiles. Collar points. Bellissard-Benedetti-Gambaudo. Aggregate collared tiles into large patches.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Other techniques

Various tricks to collar as little as possible. Barge-Diamond-Hunton-Sadun. Don’t collar tiles. Collar points. Bellissard-Benedetti-Gambaudo. Aggregate collared tiles into large patches. Can express tilings with infinite local complexity as inverse limits, too. Details depend on setting.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Table of Contents 1

Tilings

2

Types of tilings Hierarchical tilings Cut and project tilings Local matching rules

3

Local topology of tiling spaces

4

Inverse limits

5

Cohomology

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Algebraic invariants of tiling spaces

Homology groups H∗ and homotopy groups πn fail.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Algebraic invariants of tiling spaces

Homology groups H∗ and homotopy groups πn fail. So do simplicial, singular and cellular cohomology.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Algebraic invariants of tiling spaces

Homology groups H∗ and homotopy groups πn fail. So do simplicial, singular and cellular cohomology. Problem: Ω is connected but not path-connected, and path components are contractible.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Algebraic invariants of tiling spaces

Homology groups H∗ and homotopy groups πn fail. So do simplicial, singular and cellular cohomology. Problem: Ω is connected but not path-connected, and path components are contractible. ˇ Solution: Cech cohomology.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

ˇ Cech cohomology

Complicated definition involving combinatorics of open covers.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

ˇ Cech cohomology

Complicated definition involving combinatorics of open covers. Don’t need the definition! Just 2 key properties: ˇ ∗ (X ) = H ∗ (X ). If X is a CW complex, H ˇ ∗ (X ) = lim H ˇ ∗ (Xi ). If X = lim Xi , H ← − − →

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

ˇ Cech cohomology

Complicated definition involving combinatorics of open covers. Don’t need the definition! Just 2 key properties: ˇ ∗ (X ) = H ∗ (X ). If X is a CW complex, H ˇ ∗ (X ) = lim H ˇ ∗ (Xi ). If X = lim Xi , H ← − − → Strategy: Write tiling space Ω as inverse limit of CW complexes Γi . Then

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

ˇ Cech cohomology

Complicated definition involving combinatorics of open covers. Don’t need the definition! Just 2 key properties: ˇ ∗ (X ) = H ∗ (X ). If X is a CW complex, H ˇ ∗ (X ) = lim H ˇ ∗ (Xi ). If X = lim Xi , H ← − − → Strategy: Write tiling space Ω as inverse limit of CW complexes Γi . Then ˇ ∗ (Ω) = lim H ˇ ∗ (Γi ) = lim H ∗ (Γi ). H − → − →

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

ˇ Cech cohomology

Complicated definition involving combinatorics of open covers. Don’t need the definition! Just 2 key properties: ˇ ∗ (X ) = H ∗ (X ). If X is a CW complex, H ˇ ∗ (X ) = lim H ˇ ∗ (Xi ). If X = lim Xi , H ← − − → Strategy: Write tiling space Ω as inverse limit of CW complexes Γi . Then ˇ ∗ (Ω) = lim H ˇ ∗ (Γi ) = lim H ∗ (Γi ). H − → − → But we already did that!

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

The upshot

ˇ ∗ (Ω) = lim H ∗ (Γn ) H − → n→∞

is the limit of short-range cohomological information.

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Challenges

How to get at cohomology directly, without reference to specific inverse limit structure?

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Challenges

How to get at cohomology directly, without reference to specific inverse limit structure? What does it tell us about tilings?

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Challenges

How to get at cohomology directly, without reference to specific inverse limit structure? What does it tell us about tilings? What does it tell us about tiling spaces?

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Challenges

How to get at cohomology directly, without reference to specific inverse limit structure? What does it tell us about tilings? What does it tell us about tiling spaces? What does it tell us about maps between tiling spaces?

Lorenzo Sadun

Tiling Spaces and Topology

Tilings Types of tilings Local topology of tiling spaces Inverse limits Cohomology

Challenges

How to get at cohomology directly, without reference to specific inverse limit structure? What does it tell us about tilings? What does it tell us about tiling spaces? What does it tell us about maps between tiling spaces? Lots of answers. Tomorrow.

Lorenzo Sadun

Tiling Spaces and Topology