Cohomology Seminar Algorithms Jari de Kroon Eindhoven University - - PowerPoint PPT Presentation

Cohomology

Seminar Algorithms Jari de Kroon

Eindhoven University of Technology

May 22, 2018

Content

◮ Motivation ◮ Cohomology groups ◮ Universal Coefficient Theorem ◮ Application [SMV11]

Motivation

◮ Distinguish topological spaces ◮ Algebraic dual of homology

P-cochain

Recall:

For simplicial complex K, a p-chain is a formal sum c = niσi of p-simplices. Cp = Cp(K; G) is the group of p-chains closed under addition, with constants ni ∈ G.

P-cochain

Recall:

For simplicial complex K, a p-chain is a formal sum c = niσi of p-simplices. Cp = Cp(K; G) is the group of p-chains closed under addition, with constants ni ∈ G. Define a p-cochain as a homomorphism ϕ : Cp → G. These form the group of p-cochains, C p = Hom(Cp, G).

P-cochain

Recall:

For simplicial complex K, a p-chain is a formal sum c = niσi of p-simplices. Cp = Cp(K; G) is the group of p-chains closed under addition, with constants ni ∈ G. Define a p-cochain as a homomorphism ϕ : Cp → G. These form the group of p-cochains, C p = Hom(Cp, G).

Example:

Take G = Z2, then given a p-chain c ∈ Cp, the cochain maps c to 0 or 1. Intuitively can be seen as functions from p-simplices to 0 or 1, as the p-simplices form the basis of Cp.

P-cochain

The basis of Cp are the p-simplices σ1, σ2, .., σn. The basis of C p are the cochains f1, f2, ..., fn such that fi(σj) = δi,j, the Kronecker delta that is 1 if i = j.

P-cochain

The basis of Cp are the p-simplices σ1, σ2, .., σn. The basis of C p are the cochains f1, f2, ..., fn such that fi(σj) = δi,j, the Kronecker delta that is 1 if i = j.

Example:

Take ϕ(x) =

• 1,

if x = v0 or x = v1 0,

• therwise

P-cochain

The basis of Cp are the p-simplices σ1, σ2, .., σn. The basis of C p are the cochains f1, f2, ..., fn such that fi(σj) = δi,j, the Kronecker delta that is 1 if i = j.

Example:

Take ϕ(x) =

• 1,

if x = v0 or x = v1 0,

• therwise

Then ϕ(x) = f0(x) + f1(x).

Coboundary map

Recall:

The boundary operator is a homomorphism ∂p : Cp → Cp−1. For p-simplex σ = [v0, .., vp], we had ∂pσ =

p

• i=0

(−1)i[v0, .., ˆ vi, .., vp]

Coboundary map

Recall:

The boundary operator is a homomorphism ∂p : Cp → Cp−1. For p-simplex σ = [v0, .., vp], we had ∂pσ =

p

• i=0

(−1)i[v0, .., ˆ vi, .., vp] Define the coboundary operator as δp : C p → C p+1. Then given a p-cochain ϕ ∈ C p = Hom(Ci, G), we now define δpϕ([v0, .., vp+1]) =

p+1

• i=0

(−1)iϕ([v0, .., ˆ vi, .., vp+1])

Example

Take G = Z2. Let ϕ be a 0-cochain defined as ϕ(x) =

• 1,

if x = v0 0,

• therwise

Then δ0ϕ is a cochain on the edges.

v0 v1 v2

δ0ϕ([v0v1])

Example

Take G = Z2. Let ϕ be a 0-cochain defined as ϕ(x) =

• 1,

if x = v0 0,

• therwise

Then δ0ϕ is a cochain on the edges.

v0 v1 v2

δ0ϕ([v0v1]) = ϕ([v1])−ϕ([v0]) = 0−1

Example

Take G = Z2. Let ϕ be a 0-cochain defined as ϕ(x) =

• 1,

if x = v0 0,

• therwise

Then δ0ϕ is a cochain on the edges.

v0 v1 v2

δ0ϕ([v0v1]) = ϕ([v1])−ϕ([v0]) = 0−1 δ0ϕ([v1v2])

Example

Take G = Z2. Let ϕ be a 0-cochain defined as ϕ(x) =

• 1,

if x = v0 0,

• therwise

Then δ0ϕ is a cochain on the edges.

v0 v1 v2

δ0ϕ([v0v1]) = ϕ([v1])−ϕ([v0]) = 0−1 δ0ϕ([v1v2]) = ϕ([v2])−ϕ([v1]) = 0−0

Example

Take G = Z2. Let ϕ be a 0-cochain defined as ϕ(x) =

• 1,

if x = v0 0,

• therwise

Then δ0ϕ is a cochain on the edges.

v0 v1 v2

δ0ϕ([v0v1]) = ϕ([v1])−ϕ([v0]) = 0−1 δ0ϕ([v1v2]) = ϕ([v2])−ϕ([v1]) = 0−0 δ0ϕ([v2v0])

Example

Take G = Z2. Let ϕ be a 0-cochain defined as ϕ(x) =

• 1,

if x = v0 0,

• therwise

Then δ0ϕ is a cochain on the edges.

v0 v1 v2

δ0ϕ([v0v1]) = ϕ([v1])−ϕ([v0]) = 0−1 δ0ϕ([v1v2]) = ϕ([v2])−ϕ([v1]) = 0−0 δ0ϕ([v2v0]) = ϕ([v0])−ϕ([v2]) = 1−0

Example

Take G = Z2. Let ϕ be a 0-cochain defined as ϕ(x) =

• 1,

if x = v0 0,

• therwise

Then δ0ϕ is a cochain on the edges.

v0 v1 v2

δ0ϕ([v0v1]) = ϕ([v1])−ϕ([v0]) = 0−1 δ0ϕ([v1v2]) = ϕ([v2])−ϕ([v1]) = 0−0 δ0ϕ([v2v0]) = ϕ([v0])−ϕ([v2]) = 1−0

Example

Then δ1(δ0ϕ) is a cochain on the triangles.

v0 v1 v2

δ1(δ0ϕ)([v0v1v2])

Example

Then δ1(δ0ϕ) is a cochain on the triangles.

v0 v1 v2

δ1(δ0ϕ)([v0v1v2]) = δ0ϕ([v1v2]) − δ0ϕ([v0v2]) + δ0ϕ([v0v1]) = 0 − 1 + 1 = 0

Example

Then δ1(δ0ϕ) is a cochain on the triangles.

v0 v1 v2

δ1(δ0ϕ)([v0v1v2]) = δ0ϕ([v1v2]) − δ0ϕ([v0v2]) + δ0ϕ([v0v1]) = 0 − 1 + 1 = 0 No coincidence, in general δp+1δp = 0.

Example

Then δ1(δ0ϕ) is a cochain on the triangles.

v0 v1 v2

δ1(δ0ϕ)([v0v1v2]) = δ0ϕ([v1v2]) − δ0ϕ([v0v2]) + δ0ϕ([v0v1]) = 0 − 1 + 1 = 0 No coincidence, in general δp+1δp = 0. In Z2, an edge is evaluated to 1 iff vertices are labeled differently. A triangle is evaluated to 1 iff an odd number of edges evaluate to 1.

Homology groups

Recall:

Group of p-cycles Zp is the kernel of ∂p. Group of p-boundaries Bp is the image of ∂p+1. The p-th homology group Hp = Zp/Bp.

Cohomology groups

In cohomology we define: Group of p-cocycles Z p is the kernel of δp. Group of p-coboundaries Bp is the image of δp−1. The p-th cohomology group Hp = Z p/Bp.

• Ex. Homology vs cohomology groups

0 → C 0 δ0 − → C 1 δ1 − → C 2 δ2 − → 0, G = Z2

• Ex. Homology vs cohomology groups

0 → C 0 δ0 − → C 1 δ1 − → C 2 δ2 − → 0, G = Z2 rank H0 =

• Ex. Homology vs cohomology groups

0 → C 0 δ0 − → C 1 δ1 − → C 2 δ2 − → 0, G = Z2 rank H0 = 1 (connected components)

• Ex. Homology vs cohomology groups

0 → C 0 δ0 − → C 1 δ1 − → C 2 δ2 − → 0, G = Z2 0-cocycles := labeling of vertices such that all edges evaluate to zero.

• Ex. Homology vs cohomology groups

0 → C 0 δ0 − → C 1 δ1 − → C 2 δ2 − → 0, G = Z2 0-cocycles := labeling of vertices such that all edges evaluate to zero. ϕ(x) = 1

• Ex. Homology vs cohomology groups

0 → C 0 δ0 − → C 1 δ1 − → C 2 δ2 − → 0, G = Z2 0-cocycles := labeling of vertices such that all edges evaluate to zero. ϕ(x) = 1 Zero-homomorphism is trivial solution, not counted.

• Ex. Homology vs cohomology groups

0 → C 0 δ0 − → C 1 δ1 − → C 2 δ2 − → 0, G = Z2 0-cocycles := labeling of vertices such that all edges evaluate to zero. ϕ(x) = 1 Zero-homomorphism is trivial solution, not counted. 0-coboundaries do not exist, as there is no C −1.

• Ex. Homology vs cohomology groups

0 → C 0 δ0 − → C 1 δ1 − → C 2 δ2 − → 0, G = Z2 0-cocycles := labeling of vertices such that all edges evaluate to zero. ϕ(x) = 1 Zero-homomorphism is trivial solution, not counted. 0-coboundaries do not exist, as there is no C −1. So rank H0 = 1

• Ex. Homology vs cohomology groups

0 → C 0 δ0 − → C 1 δ1 − → C 2 δ2 − → 0, G = Z2 rank H1 =

• Ex. Homology vs cohomology groups

0 → C 0 δ0 − → C 1 δ1 − → C 2 δ2 − → 0, G = Z2 rank H1 = 1 (1-dimensional hole)

• Ex. Homology vs cohomology groups

0 → C 0 δ0 − → C 1 δ1 − → C 2 δ2 − → 0, G = Z2 1-cocycles := labeling of edges such that all triangles evaluate to zero.

• Ex. Homology vs cohomology groups

0 → C 0 δ0 − → C 1 δ1 − → C 2 δ2 − → 0, G = Z2 1-cocycles := labeling of edges such that all triangles evaluate to zero. Every triangle incident to even number of edges evaluating to one.

• Ex. Homology vs cohomology groups

0 → C 0 δ0 − → C 1 δ1 − → C 2 δ2 − → 0, G = Z2 1-cocycles := labeling of edges such that all triangles evaluate to zero. Every triangle incident to even number of edges evaluating to one. 1-cocycle looks like picket fence (red).

• Ex. Homology vs cohomology groups

0 → C 0 δ0 − → C 1 δ1 − → C 2 δ2 − → 0, G = Z2 1-cocycles := labeling of edges such that all triangles evaluate to zero. Every triangle incident to even number of edges evaluating to one. 1-cocycle looks like picket fence (red). 1-coboundary := coboundary of orange vertices are red edges.

• Ex. Homology vs cohomology groups

0 → C 0 δ0 − → C 1 δ1 − → C 2 δ2 − → 0, G = Z2 1-cocycles := labeling of edges such that all triangles evaluate to zero. Every triangle incident to even number of edges evaluating to one. 1-cocycle looks like picket fence (red). 1-coboundary := coboundary of orange vertices are red edges. Since Hp = Z p/Bp, we want 1-cocycles that is not also a 1-coboundary.

• Ex. Homology vs cohomology groups

0 → C 0 δ0 − → C 1 δ1 − → C 2 δ2 − → 0, G = Z2 Picket fence starting on outer ring, ending

• n inner ring.

• Ex. Homology vs cohomology groups

0 → C 0 δ0 − → C 1 δ1 − → C 2 δ2 − → 0, G = Z2 Picket fence starting on outer ring, ending

• n inner ring.

All such fences cohomologous, that is for 1-cocycles α and β, α − β is a 1-coboundary.

• Ex. Homology vs cohomology groups

0 → C 0 δ0 − → C 1 δ1 − → C 2 δ2 − → 0, G = Z2 Picket fence starting on outer ring, ending

• n inner ring.

All such fences cohomologous, that is for 1-cocycles α and β, α − β is a 1-coboundary. It follows that rank H1 = 1.

• Ex. Homology vs cohomology groups

rank Hp = rank Hp

Rank Hp = rank Zp - rank Bp.

rank Hp = rank Hp

Rank Hp = rank Zp - rank Bp. Similarly rank Hp = rank Z p - rank Bp.

rank Hp = rank Hp

Rank Hp = rank Zp - rank Bp. Similarly rank Hp = rank Z p - rank Bp. Take another look at matrices.

rank Hp = rank Hp

Rank Hp = rank Zp - rank Bp. Similarly rank Hp = rank Z p - rank Bp. Take another look at matrices.

z y x

M′ =     xy yz zx x −1 1 y 1 −1 z 1 −1     M′ is the 1-boundary matrix.

rank Hp = rank Hp

Rank Hp = rank Zp - rank Bp. Similarly rank Hp = rank Z p - rank Bp. Take another look at matrices.

z y x

M =     xy yz zx x 1 1 y 1 1 z 1 1     M is the 1-boundary matrix with coefficients in Z2 .

rank Hp = rank Hp

Lemma

A cochain evaluates a single p-simplex to one and all others to zero if and only if its coboundary evaluates each (p + 1)-coface of this p-simplex to one and all other (p + 1)-simplices to zero.

rank Hp = rank Hp

Now lets take a look at the 0-coboundary matrix N.

rank Hp = rank Hp

Now lets take a look at the 0-coboundary matrix N.

z y x

M =     xy yz zx x 1 1 y 1 1 z 1 1     M is the 1-boundary matrix.

rank Hp = rank Hp

Now lets take a look at the 0-coboundary matrix N.

z y x

M =     xy yz zx x 1 1 y 1 1 z 1 1     M is the 1-boundary matrix. N =     fx fy fz xy δfx(xy) δfy(xy) δfz(xy) yz δfx(yz) δfy(yz) δfz(yz) zx δfx(zx) δfy(zx) δfz(zx)    

rank Hp = rank Hp

Now lets take a look at the 0-coboundary matrix N.

z y x

M =     xy yz zx x 1 1 y 1 1 z 1 1     N =     fx fy fz xy 1 1 yz 1 1 zx 1 1     The 0-coboundary matrix is the transpose of the 1-boundary matrix.

rank Hp = rank Hp

The p-th and (p + 1)-st boundary matrices in normal form are also the (p − 1)-st and p-th coboundary matrices in normal form

• transposed. Since rank Bp+ rank Zp = rank Bp+ rank Z p, it

follows rank Hp = rank Hp.

rank Hp = rank Hp

The p-th and (p + 1)-st boundary matrices in normal form are also the (p − 1)-st and p-th coboundary matrices in normal form

• transposed. Since rank Bp+ rank Zp = rank Bp+ rank Z p, it

follows rank Hp = rank Hp. No notion of co-Betti numbers.

Universal Coefficient Theorem

Given a topological space, X, there are maps Hp(X) → Hom(Hp(X), G) → Hp(X) in which the first map is a natural isomorphism and the second is an isomorphism that is not natural.

Application

Nonlinear dimensionality reduction

Represent high-dimensional nonlinear data in terms of low-dimensional coordinates.

Application

Nonlinear dimensionality reduction

Represent high-dimensional nonlinear data in terms of low-dimensional coordinates.

Existing techniques:

◮ Isomaps ◮ Laplacian eigenmaps ◮ LLE

Application

Techniques struggle with circular functions, cannot be embedded in lower dimensions.

[SMV11] algorithm

Goal

Identify candidates for significant circle-structures in data using persistent cohomology.

[SMV11] algorithm

Goal

Identify candidates for significant circle-structures in data using persistent cohomology.

Algorithm steps

• 1. Represent data set X as simplicial complex (Vietoris-Rips or

Witness)

[SMV11] algorithm

Goal

Identify candidates for significant circle-structures in data using persistent cohomology.

Algorithm steps

• 1. Represent data set X as simplicial complex (Vietoris-Rips or

Witness)

• 2. Use persistent cohomology to identify significant cohomology

class αp in data. This uses a finite group Fp for some prime p.

[SMV11] algorithm

Goal

Identify candidates for significant circle-structures in data using persistent cohomology.

Algorithm steps

• 1. Represent data set X as simplicial complex (Vietoris-Rips or

Witness)

• 2. Use persistent cohomology to identify significant cohomology

class αp in data. This uses a finite group Fp for some prime p.

• 3. Lift αp to a cohomology class α with integer coefficients.

[SMV11] algorithm

Goal

Identify candidates for significant circle-structures in data using persistent cohomology.

Algorithm steps

• 1. Represent data set X as simplicial complex (Vietoris-Rips or

Witness)

• 2. Use persistent cohomology to identify significant cohomology

class αp in data. This uses a finite group Fp for some prime p.

• 3. Lift αp to a cohomology class α with integer coefficients.
• 4. Replace the integer cocycle α by a harmonic cycle α in the

same cohomology class (smoothest cocycle cohomologous to α).

[SMV11] algorithm

Goal

Identify candidates for significant circle-structures in data using persistent cohomology.

Algorithm steps

• 1. Represent data set X as simplicial complex (Vietoris-Rips or

Witness)

• 2. Use persistent cohomology to identify significant cohomology

class αp in data. This uses a finite group Fp for some prime p.

• 3. Lift αp to a cohomology class α with integer coefficients.
• 4. Replace the integer cocycle α by a harmonic cycle α in the

same cohomology class (smoothest cocycle cohomologous to α).

• 5. Integrate α to a circle-valued function θ : X → S1.

[SMV11] algorithm

[SMV11] algorithm

[SMV11] algorithm

Summary

◮ Cohomology is the dual of homology ◮ Intuitive examples with coefficients in Z2 ◮ Algorithm for finding circle functions in topological data

Sources

◮ [Hatch] A. Hatcher, Algebraic Topology. ◮ [EH10] H. Edelsbrunner, J. L. Harer, Computational topology. An introduction. American Mathematical Society, Providence, RI, 2010. ◮ [SMV11] V. de Silva, D. Morozov, and M. Vejdemo-Johansson. Persistent cohomology and circular coordinates, Discrete & Computational Geometry 45.4, 737-759, 2011.