Homology theory

Teacher: Clément Maria

\text{Simplicial Complexes } \ni 𝒦 ⟼ \lbrace H_p(𝒦) \text{ vect. space of } \dim = p ∈ ℕ \rbrace

“Connectivity” of the underlying space

  • connected component
  • holes (as in the torus)
  • voids (as in the sphere)

Data analysis:

\text{Data } ⟼ P ⊆ ℝ^D
  • Cluster algorithms:

    • with most clustering aglorithms: you get to split the data points “linearly”, in a way
    • with topology: we’ll be able to take into account the geometry of the data

Fix a field of coefficients $𝔽$ (usually: either $ℤ/p$ (very convenient for computational methods) or $ℚ$)

A chain complex $𝒞$:

is a family of $𝔽$-vector spaces $\lbrace C_p \rbrace_{p ∈ ℕ}$ and linear maps

\partial_p : C_p ⟶ C_{p-1}

satisfying

\partial_{p-1} \circ \partial_{p} = 0 \qquad ∀ p ∈ ℕ

NB: with $ℤ$ instead of $𝔽$, we have the same definition for $ℤ$-modules, that is, abelian groups

Fix $𝔽 = ℤ/2ℤ$

Ex: Let $K$ be a simplicial complex. For all integer $p ∈ ℕ$, $K_p$ will denote the simplices of dimension $p$.

We define a chain complex $𝒞(K)$:

  • C_p(K) = ⟨σ⟩_{σ ∈ K_p} the free vector space of $p$-chains with \dim C_p(K) = card K_p

  • $\partial_p$ boundary maps: for $σ ∈ K_p$

    \partial_p σ = \sum\limits_{ τ ⊆ σ, τ ∈ K_{p-1}} τ

    ex: \partial_p (σ_1 \underbrace{+}_{\text{ in } C_p} σ_2) = \partial_p (σ_1) \underbrace{+}_{\text{in } C_{p-1}} \partial_p (σ_2)

Ex: cf. picture

Lemma: \partial_{p-1} \partial_p = 0

Let $σ \; ≝ \; \lbrace v_0, …, v_p \rbrace, \quad \dim σ = p$

\partial_{p-1} \partial_p σ = \partial_{p-1} \Big( \sum\limits_{ τ ⊆ σ , \dim τ = p-1} τ \Big) = \partial_{p-1} \Big( \sum\limits_{ i=0}^p \lbrace v_0, …, \underbrace{\widehat v_i}_{\text{removed from the set}}, …, v_p\rbrace\Big) \\ = \sum\limits_{ i=0}^p \Big( \partial_{p-1} \lbrace v_0, …, \widehat v_i, …, v_p\rbrace\Big) = \sum\limits_{ i=0}^p \Big( \sum\limits_{ j<i } \lbrace v_0, …, \widehat v_j, …, \widehat v_i, …, v_p\rbrace + \sum\limits_{ j>i } \lbrace v_0, …, \widehat v_i, …, \widehat v_j, …, v_p\rbrace\Big) \\ = 0

NB: we could have done that in any characteristic, by tracking the signs of the simplices.

Example: cf. picture



Let $𝒞$ be a chain complex

\lbrace C_p, \partial_p: C_p ⟶ C_{p-1}\rbrace_{p ∈ ℕ}
$p$-cycles:
ker \partial_p \; ≝ \; Z_p ⊆ C_p
$p$-boundaries:
im \partial_{p+1} \; ≝ \; B_p ⊆ C_p

NB: these are kernels and images of linear maps, hence vector spaces

Since $\partial_p \partial_{p+1} = 0$, it comes that

im \partial_{p+1} = B_p ⊆ Z_p = ker \partial_p
$p$-homology group:
H_p = Z_p/B_p

Recall that, in the quotient, which is a vector space too (called “homology group” for historical reasons):

c \sim c' ⟺ ∃ b ∈ B_p; c = c' + b

So in $Z_p/B_p \; = \; Z_p/\sim \; = \; H_p$:

[c] = c + B_p = \lbrace c+b \; \mid \; b ∈ B_p\rbrace

Two ideas (cf. pictures with torus and annulus):

  • cycles are the objects of interest
  • those that are contractible don’t matter

The kernels or boundary maps are exactly the way to capture cycles (the objects of interest)

Homology group = abelianization of the homotopy groups (we loose the notion of direction)

Ex (for a sphere):

In 2D:

  • $\dim H_0 = 1$
  • $\dim H_1 = 0$
  • $\dim H_2 = 1$

In 3D:

  • $\dim H_0 = 1$
  • $\dim H_1 = 0$
  • $\dim H_2 = 0$
  • $\dim H_3 = 1$

Poincaré conjecture:

  • at first: « Is it the case that every 3D manifold that has the same homology groups as the sphere is a sphere ?» ⟶ No (answered by Poincaré)

  • then, stronger: « Is it the case that every 3D manifold that has the same homotopy groups as the sphere is a sphere ?» ⟶ Yes (answered by Perelmann)

Computation

$p$-th Betty number:
β_p \; ≝ \; \dim H_p = \dim Z_p - \dim B_p

NB: here, $H_p$ is a vector space so it makes sense. It would be more complicated if it were just a group.

Example: cf. picture (computation of Smith normal forms of the partial maps)

Euler characteristic

Euler characteristic:
χ(K) = \sum\limits_{ p ∈ ℕ } (-1)^p n_p \qquad \text{ where } n_p = card K_p

Claim:

χ(K) = \sum\limits_{ p ∈ ℕ } (-1)^p β_p

Proof:

β_p = \dim Z_p - \dim B_p = \dim Ker \partial_p - \dim Im \partial_{p+1}
\partial_p: C_p ⟶ C_{p-1}
\dim C_p = n_p = \dim Ker \partial_p + \dim Im \partial_p = \dim Z_p + \dim B_{p-1}

So

\sum\limits_{ p ∈ ℕ } (-1)^p β_p = \sum\limits_{ p ∈ ℕ } (-1)^p \dim Z_p + \underbrace{\sum\limits_{ p ∈ ℕ } (-1)^{p+1} \dim B_p}_{= \sum\limits_{ p ≥ 1 } (-1)^p \dim B_{p-1} = \sum\limits_{ p ∈ ℕ } (-1)^p \dim B_{p-1}} \\ = \sum\limits_{ p ∈ ℕ } (-1)^p (\underbrace{\dim Z_p + \dim B_{p-1}}_{= \; n_p})

Ex: Prove that β_0 = \# \text{connected components}

u \sim v ⟺ \text{ there exists a } (u,v)\text{-path}

$⟸$: if there exists a $(u,v)$-path ($u = u_1, …, u_n = v$)

$1$-chain (that is, a formal sum of edges)

P = u_1 u_2 + u_2 u_3 + ⋯ + u_{n-1} u_n\\ \partial_1 P = u + v

$⟹$: If $c$ is a $1$-chain, then $\partial_1 c$ has an even number of vertices. So if there exists $c$ such that

\partial c = u + v

there can’t exist $c’$ such that

\partial c' = u

(which would happen if $u$ and $v$ were in two different connected components)


Th: Let $K, K’$ be simplicial complexes st

\vert K \vert ≅ \vert K' \vert \qquad \text{ homeomorphic / homotopy eq. }

Then $∀p ∈ ℕ$:

H_p(K) ≅ H_p(K') \quad / \quad β_p(K) ≅ β_p(K')

Elementary collapses

In a simplicial complex:

a free pair $(τ, σ)$:

is such that

  • $τ, σ$ are simplices: $τ, σ ∈ K$
  • $\dim σ = \dim τ +1$
  • $σ$ is the only coface of $τ$ (that is: $τ ⊊ σ$ (the dual of a subface))

(cf. picture)

NB: this implies that $σ$ has no coface (i.e. is the subface of nothing else), otherwise: any coface of $σ$ is a coface of $τ$ too (and we don’t have uniqueness)

Removing a free pair

K ⟼ K \backslash \lbrace τ, σ\rbrace

is an elementary collapse.


Thm: Elementary collapses preserve homology:

H_p(K) ≅ H_p(K \backslash \lbrace τ, σ \rbrace) \text{ when } (τ, σ) \text{ is a free pair}

Proof: cf. picture



In general, with abelian groups:

H(K, ℤ) ≅ ℤ^{β_p} \bigoplus_{i=1}^n ℤ/q^{k_i} ℤ

Cf. Universal coefficient theorem

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