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
\[\partial_p : C_p ⟶ C_{p1}\]satisfying
\[\partial_{p1} \circ \partial_{p} = 0 \qquad ∀ p ∈ ℕ\]
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 $σ ∈ K_p$
\[\partial_p σ = \sum\limits_{ τ ⊆ σ, τ ∈ K_{p1}} τ\]ex: \(\partial_p (σ_1 \underbrace{+}_{\text{ in } C_p} σ_2) = \partial_p (σ_1) \underbrace{+}_{\text{in } C_{p1}} \partial_p (σ_2)\)
Ex: cf. picture
Lemma: \(\partial_{p1} \partial_p = 0\)
Let $σ \; ≝ \; \lbrace v_0, …, v_p \rbrace, \quad \dim σ = p$
\[\partial_{p1} \partial_p σ = \partial_{p1} \Big( \sum\limits_{ τ ⊆ σ , \dim τ = p1} τ \Big) = \partial_{p1} \Big( \sum\limits_{ i=0}^p \lbrace v_0, …, \underbrace{\widehat v_i}_{\text{removed from the set}}, …, v_p\rbrace\Big) \\ = \sum\limits_{ i=0}^p \Big( \partial_{p1} \lbrace v_0, …, \widehat v_i, …, v_p\rbrace\Big) = \sum\limits_{ i=0}^p \Big( \sum\limits_{ j<i } \lbrace v_0, …, \widehat v_j, …, \widehat v_i, …, v_p\rbrace + \sum\limits_{ j>i } \lbrace v_0, …, \widehat v_i, …, \widehat v_j, …, v_p\rbrace\Big) \\ = 0\]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
\[\lbrace C_p, \partial_p: C_p ⟶ C_{p1}\rbrace_{p ∈ ℕ}\] $p$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:
\[β_p = \dim Z_p  \dim B_p = \dim Ker \partial_p  \dim Im \partial_{p+1}\] \[\partial_p: C_p ⟶ C_{p1}\] \[\dim C_p = n_p = \dim Ker \partial_p + \dim Im \partial_p = \dim Z_p + \dim B_{p1}\]So
\[\sum\limits_{ p ∈ ℕ } (1)^p β_p = \sum\limits_{ p ∈ ℕ } (1)^p \dim Z_p + \underbrace{\sum\limits_{ p ∈ ℕ } (1)^{p+1} \dim B_p}_{= \sum\limits_{ p ≥ 1 } (1)^p \dim B_{p1} = \sum\limits_{ p ∈ ℕ } (1)^p \dim B_{p1}} \\ = \sum\limits_{ p ∈ ℕ } (1)^p (\underbrace{\dim Z_p + \dim B_{p1}}_{= \; n_p})\]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)
\[P = u_1 u_2 + u_2 u_3 + ⋯ + u_{n1} u_n\\ \partial_1 P = u + v\]$⟹$: 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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