CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Tiling Spaces and Cohomology

Lorenzo Sadun

Trondheim, June 9, 2015

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Outline

1 Cohomology

2 Examples

3 Pattern EquivarianceBut what does PE cohomology mean?

4 Understanding H1(Ω,Z) and H1(Ω,Rd).

5 Maps between (FLC) tiling spaces

6 Image of H1 under the Ruelle-Sullivan map

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Outline

1 Cohomology

2 Examples

3 Pattern EquivarianceBut what does PE cohomology mean?

4 Understanding H1(Ω,Z) and H1(Ω,Rd).

5 Maps between (FLC) tiling spaces

6 Image of H1 under the Ruelle-Sullivan map

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Outline

1 Cohomology

2 Examples

3 Pattern EquivarianceBut what does PE cohomology mean?

4 Understanding H1(Ω,Z) and H1(Ω,Rd).

5 Maps between (FLC) tiling spaces

6 Image of H1 under the Ruelle-Sullivan map

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Outline

1 Cohomology

2 Examples

3 Pattern EquivarianceBut what does PE cohomology mean?

4 Understanding H1(Ω,Z) and H1(Ω,Rd).

5 Maps between (FLC) tiling spaces

6 Image of H1 under the Ruelle-Sullivan map

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Outline

1 Cohomology

2 Examples

3 Pattern EquivarianceBut what does PE cohomology mean?

4 Understanding H1(Ω,Z) and H1(Ω,Rd).

5 Maps between (FLC) tiling spaces

6 Image of H1 under the Ruelle-Sullivan map

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Outline

1 Cohomology

2 Examples

3 Pattern EquivarianceBut what does PE cohomology mean?

4 Understanding H1(Ω,Z) and H1(Ω,Rd).

5 Maps between (FLC) tiling spaces

6 Image of H1 under the Ruelle-Sullivan map

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Table of Contents

1 Cohomology

2 Examples

3 Pattern EquivarianceBut what does PE cohomology mean?

4 Understanding H1(Ω,Z) and H1(Ω,Rd).

5 Maps between (FLC) tiling spaces

6 Image of H1 under the Ruelle-Sullivan map

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

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 pathcomponents are contractible.

Solution: Cech cohomology.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

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 pathcomponents are contractible.

Solution: Cech cohomology.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

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 pathcomponents are contractible.

Solution: Cech cohomology.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

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 pathcomponents are contractible.

Solution: Cech cohomology.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Cech cohomology

Complicated definition involving combinatorics of open covers.

Don’t need the definition! Just 2 key properties:

If X is a CW complex, H∗(X ) = H∗(X ).

If X = lim←−Xi , H∗(X ) = lim−→ H∗(Xi ).

Strategy: Write tiling space Ω as inverse limit of CW complexes Γi .Then

H∗(Ω) = lim−→ H∗(Γi ) = lim−→H∗(Γi ).

But we already did that!

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Cech cohomology

Complicated definition involving combinatorics of open covers.

Don’t need the definition! Just 2 key properties:

If X is a CW complex, H∗(X ) = H∗(X ).

If X = lim←−Xi , H∗(X ) = lim−→ H∗(Xi ).

Strategy: Write tiling space Ω as inverse limit of CW complexes Γi .Then

H∗(Ω) = lim−→ H∗(Γi ) = lim−→H∗(Γi ).

But we already did that!

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Cech cohomology

Complicated definition involving combinatorics of open covers.

Don’t need the definition! Just 2 key properties:

If X is a CW complex, H∗(X ) = H∗(X ).

If X = lim←−Xi , H∗(X ) = lim−→ H∗(Xi ).

Strategy: Write tiling space Ω as inverse limit of CW complexes Γi .Then

H∗(Ω) = lim−→ H∗(Γi ) = lim−→H∗(Γi ).

But we already did that!

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Cech cohomology

Complicated definition involving combinatorics of open covers.

Don’t need the definition! Just 2 key properties:

If X is a CW complex, H∗(X ) = H∗(X ).

If X = lim←−Xi , H∗(X ) = lim−→ H∗(Xi ).

Strategy: Write tiling space Ω as inverse limit of CW complexes Γi .Then

H∗(Ω) = lim−→ H∗(Γi ) = lim−→H∗(Γi ).

But we already did that!

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Cech cohomology

Complicated definition involving combinatorics of open covers.

Don’t need the definition! Just 2 key properties:

If X is a CW complex, H∗(X ) = H∗(X ).

If X = lim←−Xi , H∗(X ) = lim−→ H∗(Xi ).

Strategy: Write tiling space Ω as inverse limit of CW complexes Γi .Then

H∗(Ω) = lim−→ H∗(Γi ) = lim−→H∗(Γi ).

But we already did that!

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

The upshot

H∗(Ω) = lim−→n→∞

H∗(Γn)

is the limit of short-range cohomological information.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Challenges

How to get at cohomology directly, without reference tospecific 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 Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Table of Contents

1 Cohomology

2 Examples

3 Pattern EquivarianceBut what does PE cohomology mean?

4 Understanding H1(Ω,Z) and H1(Ω,Rd).

5 Maps between (FLC) tiling spaces

6 Image of H1 under the Ruelle-Sullivan map

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Dyadic solenoid (not actually a tiling space)

H0(S1) = H1(S1) = Z.

Double wrap is ×2 on H1 and identity on H0.

H1(Ω) = lim(Z,×2) = Z[1/2]; H0(Ω) = Z.

Z[1/2] = m/2n.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Dyadic solenoid (not actually a tiling space)

H0(S1) = H1(S1) = Z.

Double wrap is ×2 on H1 and identity on H0.

H1(Ω) = lim(Z,×2) = Z[1/2]; H0(Ω) = Z.

Z[1/2] = m/2n.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Dyadic solenoid (not actually a tiling space)

H0(S1) = H1(S1) = Z.

Double wrap is ×2 on H1 and identity on H0.

H1(Ω) = lim(Z,×2) = Z[1/2]; H0(Ω) = Z.

Z[1/2] = m/2n.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Dyadic solenoid (not actually a tiling space)

H0(S1) = H1(S1) = Z.

Double wrap is ×2 on H1 and identity on H0.

H1(Ω) = lim(Z,×2) = Z[1/2]; H0(Ω) = Z.

Z[1/2] = m/2n.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Dyadic solenoid (not actually a tiling space)

H0(S1) = H1(S1) = Z.

Double wrap is ×2 on H1 and identity on H0.

H1(Ω) = lim(Z,×2) = Z[1/2]; H0(Ω) = Z.

Z[1/2] = m/2n.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Fibonacci

H1(Γn) = Z2; H1(Ω) = lim(Z2, ( 1 11 0 )) = Z2.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Fibonacci

H1(Γn) = Z2; H1(Ω) = lim(Z2, ( 1 11 0 )) = Z2.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Chair

Approximant has H1(Γ) = Z2, H2(Γ) = Z3.

Substitution acts as ×2 on H1, as matrix with eigenvalues4, 2, 2 on H2.

H1(Ω) = Z[1/2]2, H2(Ω) =1

3Z[1/4]⊕ Z[1/2]2.

But what does that tell us??!

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Chair

Approximant has H1(Γ) = Z2, H2(Γ) = Z3.

Substitution acts as ×2 on H1, as matrix with eigenvalues4, 2, 2 on H2.

H1(Ω) = Z[1/2]2, H2(Ω) =1

3Z[1/4]⊕ Z[1/2]2.

But what does that tell us??!

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Chair

Approximant has H1(Γ) = Z2, H2(Γ) = Z3.

Substitution acts as ×2 on H1, as matrix with eigenvalues4, 2, 2 on H2.

H1(Ω) = Z[1/2]2, H2(Ω) =1

3Z[1/4]⊕ Z[1/2]2.

But what does that tell us??!

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Chair

Approximant has H1(Γ) = Z2, H2(Γ) = Z3.

Substitution acts as ×2 on H1, as matrix with eigenvalues4, 2, 2 on H2.

H1(Ω) = Z[1/2]2, H2(Ω) =1

3Z[1/4]⊕ Z[1/2]2.

But what does that tell us??!

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Penrose

Approximant has H1(Γ) = Z5, H2(Γ) = Z8.

Substitution acts by invertible matrices on both H1 and H2.

H1(Ω) = Z5, H2(Ω) = Z8.

Again, so what??!

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Penrose

Approximant has H1(Γ) = Z5, H2(Γ) = Z8.

Substitution acts by invertible matrices on both H1 and H2.

H1(Ω) = Z5, H2(Ω) = Z8.

Again, so what??!

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Penrose

Approximant has H1(Γ) = Z5, H2(Γ) = Z8.

Substitution acts by invertible matrices on both H1 and H2.

H1(Ω) = Z5, H2(Ω) = Z8.

Again, so what??!

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Penrose

Approximant has H1(Γ) = Z5, H2(Γ) = Z8.

Substitution acts by invertible matrices on both H1 and H2.

H1(Ω) = Z5, H2(Ω) = Z8.

Again, so what??!

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

Table of Contents

1 Cohomology

2 Examples

3 Pattern EquivarianceBut what does PE cohomology mean?

4 Understanding H1(Ω,Z) and H1(Ω,Rd).

5 Maps between (FLC) tiling spaces

6 Image of H1 under the Ruelle-Sullivan map

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

Pattern equivariance

Tiling T gives CW decomposition of Rd .

0-cochain assigns numbers to vertices of T

1-cochain assigns numbers to edges of T , etc

Cochain α is strongly pattern equivariant (sPE) with radius Rif α(x) depends only on (T − x) ∩ BR(0).

The uniform limit of sPE is weakly pattern equivariant (wPE).

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

Pattern equivariance

Tiling T gives CW decomposition of Rd .

0-cochain assigns numbers to vertices of T

1-cochain assigns numbers to edges of T , etc

Cochain α is strongly pattern equivariant (sPE) with radius Rif α(x) depends only on (T − x) ∩ BR(0).

The uniform limit of sPE is weakly pattern equivariant (wPE).

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

Pattern equivariance

Tiling T gives CW decomposition of Rd .

0-cochain assigns numbers to vertices of T

1-cochain assigns numbers to edges of T , etc

Cochain α is strongly pattern equivariant (sPE) with radius Rif α(x) depends only on (T − x) ∩ BR(0).

The uniform limit of sPE is weakly pattern equivariant (wPE).

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

Pattern equivariance

Tiling T gives CW decomposition of Rd .

0-cochain assigns numbers to vertices of T

1-cochain assigns numbers to edges of T , etc

Cochain α is strongly pattern equivariant (sPE) with radius Rif α(x) depends only on (T − x) ∩ BR(0).

The uniform limit of sPE is weakly pattern equivariant (wPE).

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

PE Cohomology

HkPE (T ) =

Closed sPE k-cochains

δ(sPE (k − 1)-cochains).

Theorem

If T is a repetitive FLC tiling, then

HkPE (T ) = Hk(ΩT ).

Can also look at sPE differential forms, and get de Rham-liketheory, isomorphic to Hk(ΩT ,R). (Kellendonk-Putnam)

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

PE Cohomology

HkPE (T ) =

Closed sPE k-cochains

δ(sPE (k − 1)-cochains).

Theorem

If T is a repetitive FLC tiling, then

HkPE (T ) = Hk(ΩT ).

Can also look at sPE differential forms, and get de Rham-liketheory, isomorphic to Hk(ΩT ,R). (Kellendonk-Putnam)

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

PE Cohomology

HkPE (T ) =

Closed sPE k-cochains

δ(sPE (k − 1)-cochains).

Theorem

If T is a repetitive FLC tiling, then

HkPE (T ) = Hk(ΩT ).

Can also look at sPE differential forms, and get de Rham-liketheory, isomorphic to Hk(ΩT ,R). (Kellendonk-Putnam)

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

Proof/Hand wave

A k-cochain is sPE if and only if:

It depends on a finite piece of T − x .

It is the pullback of a cochain on an approximant Γn.

Looking at sPE cochains of arbitrary radius is like taking n→∞limit.

HPE (T ) = lim−→n→∞

H∗(Γn) = H∗(Ω)

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

Proof/Hand wave

A k-cochain is sPE if and only if:

It depends on a finite piece of T − x .

It is the pullback of a cochain on an approximant Γn.

Looking at sPE cochains of arbitrary radius is like taking n→∞limit.

HPE (T ) = lim−→n→∞

H∗(Γn) = H∗(Ω)

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

Proof/Hand wave

A k-cochain is sPE if and only if:

It depends on a finite piece of T − x .

It is the pullback of a cochain on an approximant Γn.

Looking at sPE cochains of arbitrary radius is like taking n→∞limit.

HPE (T ) = lim−→n→∞

H∗(Γn) = H∗(Ω)

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

Proof/Hand wave

A k-cochain is sPE if and only if:

It depends on a finite piece of T − x .

It is the pullback of a cochain on an approximant Γn.

Looking at sPE cochains of arbitrary radius is like taking n→∞limit.

HPE (T ) = lim−→n→∞

H∗(Γn) = H∗(Ω)

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

Proof/Hand wave

A k-cochain is sPE if and only if:

It depends on a finite piece of T − x .

It is the pullback of a cochain on an approximant Γn.

Looking at sPE cochains of arbitrary radius is like taking n→∞limit.

HPE (T ) = lim−→n→∞

H∗(Γn) = H∗(Ω)

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

Chair revisited

Rewrite in terms of arrow tiles:

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

Arrow substitution

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

Understanding H1 for chair.

For the chair tiling, H1(Ω) = Z[1/2]2.

Generator (1, 0) is cochain that is 1 on all horizontal edges.

Generator (2−n, 0) is cochain that is 1 on left-most horizontaledges of n-supertiles.

Vertical generators are similar.

H1 simply counts supertile periods in horizontal and verticaldirections.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

Understanding H1 for chair.

For the chair tiling, H1(Ω) = Z[1/2]2.

Generator (1, 0) is cochain that is 1 on all horizontal edges.

Generator (2−n, 0) is cochain that is 1 on left-most horizontaledges of n-supertiles.

Vertical generators are similar.

H1 simply counts supertile periods in horizontal and verticaldirections.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

Understanding H1 for chair.

For the chair tiling, H1(Ω) = Z[1/2]2.

Generator (1, 0) is cochain that is 1 on all horizontal edges.

Generator (2−n, 0) is cochain that is 1 on left-most horizontaledges of n-supertiles.

Vertical generators are similar.

H1 simply counts supertile periods in horizontal and verticaldirections.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

Understanding H1 for chair.

For the chair tiling, H1(Ω) = Z[1/2]2.

Generator (1, 0) is cochain that is 1 on all horizontal edges.

Generator (2−n, 0) is cochain that is 1 on left-most horizontaledges of n-supertiles.

Vertical generators are similar.

H1 simply counts supertile periods in horizontal and verticaldirections.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

Understanding H1 for chair.

For the chair tiling, H1(Ω) = Z[1/2]2.

Generator (1, 0) is cochain that is 1 on all horizontal edges.

Generator (2−n, 0) is cochain that is 1 on left-most horizontaledges of n-supertiles.

Vertical generators are similar.

H1 simply counts supertile periods in horizontal and verticaldirections.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

Understanding H2 for chair.

For the chair tiling, H2(Ω) =1

3Z[1/4]⊕ Z[1/2]2.

Generator (1, 0, 0) is 1 on all (square) tiles.

Generator (4−n, 0, 0) counts n-supertiles.

Generator (1

3, 0, 0) counts chairs.

Z[1/2]2 generators count vector sum of arrows.

With PE cochains, generators of cohomology count localfeatures.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

Understanding H2 for chair.

For the chair tiling, H2(Ω) =1

3Z[1/4]⊕ Z[1/2]2.

Generator (1, 0, 0) is 1 on all (square) tiles.

Generator (4−n, 0, 0) counts n-supertiles.

Generator (1

3, 0, 0) counts chairs.

Z[1/2]2 generators count vector sum of arrows.

With PE cochains, generators of cohomology count localfeatures.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

Understanding H2 for chair.

For the chair tiling, H2(Ω) =1

3Z[1/4]⊕ Z[1/2]2.

Generator (1, 0, 0) is 1 on all (square) tiles.

Generator (4−n, 0, 0) counts n-supertiles.

Generator (1

3, 0, 0) counts chairs.

Z[1/2]2 generators count vector sum of arrows.

With PE cochains, generators of cohomology count localfeatures.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

Understanding H2 for chair.

For the chair tiling, H2(Ω) =1

3Z[1/4]⊕ Z[1/2]2.

Generator (1, 0, 0) is 1 on all (square) tiles.

Generator (4−n, 0, 0) counts n-supertiles.

Generator (1

3, 0, 0) counts chairs.

Z[1/2]2 generators count vector sum of arrows.

With PE cochains, generators of cohomology count localfeatures.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

Understanding H2 for chair.

For the chair tiling, H2(Ω) =1

3Z[1/4]⊕ Z[1/2]2.

Generator (1, 0, 0) is 1 on all (square) tiles.

Generator (4−n, 0, 0) counts n-supertiles.

Generator (1

3, 0, 0) counts chairs.

Z[1/2]2 generators count vector sum of arrows.

With PE cochains, generators of cohomology count localfeatures.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

Understanding H2 for chair.

For the chair tiling, H2(Ω) =1

3Z[1/4]⊕ Z[1/2]2.

Generator (1, 0, 0) is 1 on all (square) tiles.

Generator (4−n, 0, 0) counts n-supertiles.

Generator (1

3, 0, 0) counts chairs.

Z[1/2]2 generators count vector sum of arrows.

With PE cochains, generators of cohomology count localfeatures.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

Ruelle-Sullivan map

Express cohomology class with differential form α.

Average α(0) over all tilings using invariant measure.

If Ω uniquely ergodic, same as averaging α(x) over x ∈ Rd .

Sends classes in Hk(Ω) to exterior algebra of Rd .

Ring homomorphism (Kellendonk-Putnam).

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

Ruelle-Sullivan map

Express cohomology class with differential form α.

Average α(0) over all tilings using invariant measure.

If Ω uniquely ergodic, same as averaging α(x) over x ∈ Rd .

Sends classes in Hk(Ω) to exterior algebra of Rd .

Ring homomorphism (Kellendonk-Putnam).

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

Ruelle-Sullivan map

Express cohomology class with differential form α.

Average α(0) over all tilings using invariant measure.

If Ω uniquely ergodic, same as averaging α(x) over x ∈ Rd .

Sends classes in Hk(Ω) to exterior algebra of Rd .

Ring homomorphism (Kellendonk-Putnam).

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

Ruelle-Sullivan map

Express cohomology class with differential form α.

Average α(0) over all tilings using invariant measure.

If Ω uniquely ergodic, same as averaging α(x) over x ∈ Rd .

Sends classes in Hk(Ω) to exterior algebra of Rd .

Ring homomorphism (Kellendonk-Putnam).

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

But what does PE cohomology mean?

Frequency module

Assume Ω uniquely ergodic. (I.e. all patches have well-definedfrequencies.)

Let P be any patch. Can make indicator cochain iP .

Ruelle-Sullivan sends [iP ] to frequency of P.

[iP ]’s generate Hd(Ω,Z). Image of Ruelle-Sullivan isfrequency module.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Table of Contents

1 Cohomology

2 Examples

3 Pattern EquivarianceBut what does PE cohomology mean?

4 Understanding H1(Ω,Z) and H1(Ω,Rd).

5 Maps between (FLC) tiling spaces

6 Image of H1 under the Ruelle-Sullivan map

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Shape deformations

Λ = vertices of T . A shape deformation moves points around in acontrolled way.

F : Λ→ Rd ; Λ′ = x + F (x)|x ∈ Λ

Preserves FLC if δF is sPE. (Always want this!)

Generates topological conjugacy iff F is wPE.

A shape conjugacy is a shape deformation that is a topologicalconjugacy.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Shape deformations

Λ = vertices of T . A shape deformation moves points around in acontrolled way.

F : Λ→ Rd ; Λ′ = x + F (x)|x ∈ Λ

Preserves FLC if δF is sPE. (Always want this!)

Generates topological conjugacy iff F is wPE.

A shape conjugacy is a shape deformation that is a topologicalconjugacy.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Shape deformations

Λ = vertices of T . A shape deformation moves points around in acontrolled way.

F : Λ→ Rd ; Λ′ = x + F (x)|x ∈ Λ

Preserves FLC if δF is sPE. (Always want this!)

Generates topological conjugacy iff F is wPE.

A shape conjugacy is a shape deformation that is a topologicalconjugacy.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Asymptotically negligible

Definition: Closed sPE 1-cochain α is asymptoticallynegligible if α = δF with F wPE.

Theorem (Gottschalk-Hedlund) A closed sPE 1-cochain isasymptotically negligible if and only if its integral is bounded.

H1an(Ω) = classes represented by asymptotically negligible

1-cochains.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Asymptotically negligible

Definition: Closed sPE 1-cochain α is asymptoticallynegligible if α = δF with F wPE.

Theorem (Gottschalk-Hedlund) A closed sPE 1-cochain isasymptotically negligible if and only if its integral is bounded.

H1an(Ω) = classes represented by asymptotically negligible

1-cochains.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Asymptotically negligible

Definition: Closed sPE 1-cochain α is asymptoticallynegligible if α = δF with F wPE.

Theorem (Gottschalk-Hedlund) A closed sPE 1-cochain isasymptotically negligible if and only if its integral is bounded.

H1an(Ω) = classes represented by asymptotically negligible

1-cochains.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

H1 classifies deformations

Theorem (Clark-S, Kellendonk)

Infinitesimal shape changes (modulo MLD) are parametrized byH1(Ω,Rd). Infinitesimal shape conjugacies (mod MLD) areparametrized by H1

an(Ω,Rd).

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Table of Contents

1 Cohomology

2 Examples

3 Pattern EquivarianceBut what does PE cohomology mean?

4 Understanding H1(Ω,Z) and H1(Ω,Rd).

5 Maps between (FLC) tiling spaces

6 Image of H1 under the Ruelle-Sullivan map

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

All conjugacies are shape conjugacies

Theorem (Kellendonk-S ’13)

Every topological conjugacy between repetitive FLC tiling spacescan be written as the composition of a shape conjugacy and anMLD map.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

An invariant of homeomorphisms

Theorem (Julien, Julien-S)

One can associate a class in H1(Ω,Rd) to every homeomorphismΩ→ Ω′. If two homeomorphisms h1,2 : Ω→ Ω1,2 have same class,then Ω1 and Ω2 are related by an MLD map.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

How many homeomorphisms exist?

Theorem (Julien-S)

If Ω is uniquely ergodic and h : Ω→ Ω′ is a homemorphism, thenthe Ruelle-Sullivan map sends the class of h to an invertiblematrix. Conversely, if [α] ∈ H1(Ω,Rd) is sent to an invertiblematrix, there is a shape change whose class is [α].

Corollary

If Ω is uniquely ergodic and Ω′ is homeomorphic to Ω, then Ω′ isMLD to a shape change of Ω.

Analogous theorems hold for tiling spaces without FLC, using“weak” cohomology.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

How many homeomorphisms exist?

Theorem (Julien-S)

If Ω is uniquely ergodic and h : Ω→ Ω′ is a homemorphism, thenthe Ruelle-Sullivan map sends the class of h to an invertiblematrix. Conversely, if [α] ∈ H1(Ω,Rd) is sent to an invertiblematrix, there is a shape change whose class is [α].

Corollary

If Ω is uniquely ergodic and Ω′ is homeomorphic to Ω, then Ω′ isMLD to a shape change of Ω.

Analogous theorems hold for tiling spaces without FLC, using“weak” cohomology.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

How many homeomorphisms exist?

Theorem (Julien-S)

If Ω is uniquely ergodic and h : Ω→ Ω′ is a homemorphism, thenthe Ruelle-Sullivan map sends the class of h to an invertiblematrix. Conversely, if [α] ∈ H1(Ω,Rd) is sent to an invertiblematrix, there is a shape change whose class is [α].

Corollary

If Ω is uniquely ergodic and Ω′ is homeomorphic to Ω, then Ω′ isMLD to a shape change of Ω.

Analogous theorems hold for tiling spaces without FLC, using“weak” cohomology.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Proofs

(On the blackboard, if time allows)

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Proofs

(On the blackboard, if time allows)

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Table of Contents

1 Cohomology

2 Examples

3 Pattern EquivarianceBut what does PE cohomology mean?

4 Understanding H1(Ω,Z) and H1(Ω,Rd).

5 Maps between (FLC) tiling spaces

6 Image of H1 under the Ruelle-Sullivan map

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

A conjecture we all believed

Conjecture (Giordano, Putnam, Skau)

The image of H1(Ω,Z) under the Ruelle-Sullivan map is dense in(Rd)∗.

Theorem (Clark-S ’15)

There exists a uniquely ergodic 2-dimensional FLC tiling space,with minimal R2 action, for which the image of H1 underRuelle-Sullivan is Z2. In fact, H1 = Z2.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

A conjecture we all believed

Conjecture (Giordano, Putnam, Skau)

The image of H1(Ω,Z) under the Ruelle-Sullivan map is dense in(Rd)∗.

Theorem (Clark-S ’15)

There exists a uniquely ergodic 2-dimensional FLC tiling space,with minimal R2 action, for which the image of H1 underRuelle-Sullivan is Z2. In fact, H1 = Z2.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Virtual eigenvalues

Definition

An element λ ∈ (Rd)∗ is a virtual eigenvalue of a tiling space Ω ifthere exists an arbitrarily small shape change Ω→ Ω′ such that λis a topological eigenvalue of Ω′.

Theorem (Clark-S ’15)

Suppose that Ω is uniquely ergodic. Then λ is a virtual eigenvalueof Ω if and only if λ is in the closure of the image of H1(Ω,Z)under Ruelle-Sullivan.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Virtual eigenvalues

Definition

An element λ ∈ (Rd)∗ is a virtual eigenvalue of a tiling space Ω ifthere exists an arbitrarily small shape change Ω→ Ω′ such that λis a topological eigenvalue of Ω′.

Theorem (Clark-S ’15)

Suppose that Ω is uniquely ergodic. Then λ is a virtual eigenvalueof Ω if and only if λ is in the closure of the image of H1(Ω,Z)under Ruelle-Sullivan.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Torus fibrations

Theorem (Clark-S ’15)

Let Ω be uniquely ergodic, let L ∈ Rd be a lattice, and letL′ ∈ (Rd)∗ be the dual lattice. The following are equivalent:

An arbitrarily small shape change turns Ω into a Cantorbundle over Rd/L.

All elements of L′ are virtual eigenvalues.

All elements of L′ are in the closure of the image of RS.

Corollary

A tiling space can be realized as a Cantor bundle over a torusRd/(RZ)d for R arbitrarily large, and with an arbitrarily smallshape change, if and only if the image of Ruelle-Sullivan is dense.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Torus fibrations

Theorem (Clark-S ’15)

Let Ω be uniquely ergodic, let L ∈ Rd be a lattice, and letL′ ∈ (Rd)∗ be the dual lattice. The following are equivalent:

An arbitrarily small shape change turns Ω into a Cantorbundle over Rd/L.

All elements of L′ are virtual eigenvalues.

All elements of L′ are in the closure of the image of RS.

Corollary

A tiling space can be realized as a Cantor bundle over a torusRd/(RZ)d for R arbitrarily large, and with an arbitrarily smallshape change, if and only if the image of Ruelle-Sullivan is dense.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

The counterexample (Frank-Robinson DPV)

Four tiles a, b, c , d . All unit square. Build into supertiles withfusion rules:

Pn+1(a) =

Pn(b) Pn(d) Pn(d) Pn(d)Pn(c) Pn(c) Pn(a) Pn(d)Pn(d) Pn(d) Pn(b) Pn(d)Pn(d) Pn(d) Pn(b) Pn(c)

, Pn+1(c) =

Pn(b)Pn(b)Pn(b)Pn(a)

,Pn+1(b) =

[Pn(a) Pn(c) Pn(c) Pn(c)

], Pn+1(d) = [Pn(a)].

Supertiles do not meet full-edge to full-edge. All offsets arepossible. Tiling space admits shears.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

The counterexample (Frank-Robinson DPV)

Four tiles a, b, c , d . All unit square. Build into supertiles withfusion rules:

Pn+1(a) =

Pn(b) Pn(d) Pn(d) Pn(d)Pn(c) Pn(c) Pn(a) Pn(d)Pn(d) Pn(d) Pn(b) Pn(d)Pn(d) Pn(d) Pn(b) Pn(c)

, Pn+1(c) =

Pn(b)Pn(b)Pn(b)Pn(a)

,Pn+1(b) =

[Pn(a) Pn(c) Pn(c) Pn(c)

], Pn+1(d) = [Pn(a)].

Supertiles do not meet full-edge to full-edge. All offsets arepossible. Tiling space admits shears.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Shears kill off most of H1.

(Details to be shown on board, if time allows)

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Summary

Cohomology is simplest topological invariant of tiling spacesthat works.

Pattern equivariance allows us to make sense of cochains andclasses.

Top cohomology determines frequency module. Important forK-theory.

H1(Ω,Rd) parametrizes shape changes.

Up to MLD (and translations), all homeomorphisms are shapechanges.

Image of H1 under Ruelle-Sullivan controls bundle structuresand virtual eigenvalues.

Tilings with shears have very small H1.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Summary

Cohomology is simplest topological invariant of tiling spacesthat works.

Pattern equivariance allows us to make sense of cochains andclasses.

Top cohomology determines frequency module. Important forK-theory.

H1(Ω,Rd) parametrizes shape changes.

Up to MLD (and translations), all homeomorphisms are shapechanges.

Image of H1 under Ruelle-Sullivan controls bundle structuresand virtual eigenvalues.

Tilings with shears have very small H1.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Summary

Cohomology is simplest topological invariant of tiling spacesthat works.

Pattern equivariance allows us to make sense of cochains andclasses.

Top cohomology determines frequency module. Important forK-theory.

H1(Ω,Rd) parametrizes shape changes.

Up to MLD (and translations), all homeomorphisms are shapechanges.

Image of H1 under Ruelle-Sullivan controls bundle structuresand virtual eigenvalues.

Tilings with shears have very small H1.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Summary

Cohomology is simplest topological invariant of tiling spacesthat works.

Pattern equivariance allows us to make sense of cochains andclasses.

Top cohomology determines frequency module. Important forK-theory.

H1(Ω,Rd) parametrizes shape changes.

Up to MLD (and translations), all homeomorphisms are shapechanges.

Image of H1 under Ruelle-Sullivan controls bundle structuresand virtual eigenvalues.

Tilings with shears have very small H1.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Summary

Cohomology is simplest topological invariant of tiling spacesthat works.

Pattern equivariance allows us to make sense of cochains andclasses.

Top cohomology determines frequency module. Important forK-theory.

H1(Ω,Rd) parametrizes shape changes.

Up to MLD (and translations), all homeomorphisms are shapechanges.

Image of H1 under Ruelle-Sullivan controls bundle structuresand virtual eigenvalues.

Tilings with shears have very small H1.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Summary

Cohomology is simplest topological invariant of tiling spacesthat works.

Pattern equivariance allows us to make sense of cochains andclasses.

Top cohomology determines frequency module. Important forK-theory.

H1(Ω,Rd) parametrizes shape changes.

Up to MLD (and translations), all homeomorphisms are shapechanges.

Image of H1 under Ruelle-Sullivan controls bundle structuresand virtual eigenvalues.

Tilings with shears have very small H1.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Summary

Cohomology is simplest topological invariant of tiling spacesthat works.

Pattern equivariance allows us to make sense of cochains andclasses.

Top cohomology determines frequency module. Important forK-theory.

H1(Ω,Rd) parametrizes shape changes.

Up to MLD (and translations), all homeomorphisms are shapechanges.

Image of H1 under Ruelle-Sullivan controls bundle structuresand virtual eigenvalues.

Tilings with shears have very small H1.

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Open questions

If d > 2, what do H2, . . . ,Hd−1 tell us?

How much can we do without assuming unique ergodicity?

What can cohomology tell us about maps (not justhomeomorphisms) between tiling spaces?

How is classifying homeomorphisms different from classifyinghomeomorphic images? What does the group ofself-homeomorphisms of a tiling space look like?

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Open questions

If d > 2, what do H2, . . . ,Hd−1 tell us?

How much can we do without assuming unique ergodicity?

What can cohomology tell us about maps (not justhomeomorphisms) between tiling spaces?

How is classifying homeomorphisms different from classifyinghomeomorphic images? What does the group ofself-homeomorphisms of a tiling space look like?

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Open questions

If d > 2, what do H2, . . . ,Hd−1 tell us?

How much can we do without assuming unique ergodicity?

What can cohomology tell us about maps (not justhomeomorphisms) between tiling spaces?

How is classifying homeomorphisms different from classifyinghomeomorphic images? What does the group ofself-homeomorphisms of a tiling space look like?

Lorenzo Sadun Tiling Spaces and Cohomology

CohomologyExamples

Pattern EquivarianceUnderstanding H1(Ω, Z) and H1(Ω,Rd ).

Maps between (FLC) tiling spacesImage of H1 under the Ruelle-Sullivan map

Open questions

If d > 2, what do H2, . . . ,Hd−1 tell us?

How much can we do without assuming unique ergodicity?

What can cohomology tell us about maps (not justhomeomorphisms) between tiling spaces?

How is classifying homeomorphisms different from classifyinghomeomorphic images? What does the group ofself-homeomorphisms of a tiling space look like?

Lorenzo Sadun Tiling Spaces and Cohomology