Discrete Morse theory

Teacher: Clément Maria

cf. picture

Let $K$ be a simplicial complex, $<$ a partial order on $K$ st

\[τ < σ \quad ⟺ \quad \begin{cases} τ ⊆ σ \\ \dim σ = \dim τ + 1 \end{cases}\]

Boundary of $σ$:

\[\lbrace τ \; \mid \; τ < σ\rbrace\]

Coboundary of $τ$:

\[\lbrace σ \; \mid \; τ < σ\rbrace\]

The Hasse diagram of $K$:

is the directed graph $H \; ≝ \; (V, E)$ st

• $V = K$
• \[σ → τ ∈ E \quad ⟺ \quad τ < σ\]

Ex: cf. picture

A partial matching of $K$:

is a partition of the simplices \(K = X \sqcup T \sqcup S\) with a bijection $ω: S ⟶ T$ st \(ω(τ) = σ ⟹ τ < σ\)

The set $X$ is the set of critical faces.

If $K = X \sqcup T \sqcup S$, define the graph

$\overline{H} = (\overline{V}, \overline{E})$ st:

if $H \; ≝ \; (V, E)$ is the Hasse diagram:

• $\overline{V} = V = K$
• $(τ → σ)$ if $τ ∈ T, σ ∈ S$, and $ω(τ) = σ$
• $(σ → τ)$ if $(σ → τ) ∈ E$ and $(τ, σ)$ are not paired in the matching

Ex: cf. picture

NB: reverse all the edges associated to pairs that appear in the matching

If $\overline{H}$ is acyclic, the matching is said to be a Morse matching.

The Euler characteristic is preserved (number of faces of dim $0$ - number of faces of dim $1$ + number of faces of dim $2$ - ⋯).

Morse matching algorithm (Benedetti, Lutz)

# A: the set of available simplices
A ← K
Compute X ⊔ T ⊔ S : X, S, T ← ∅
while A ≠ ∅:
    if there is a free pair (τ, σ) in A:
        # τ, σ ∈ A, coboundary(τ) ∩ A = {σ}
        match τ with σ: # ω(τ) = σ
            A ← A \{τ, σ}
            T ← T ∪ {τ}
            S ← S ∪ {σ}
    else # make any maximal face σ ∈ A critical:
        A ← A \{σ}
        X ← X ∪ {σ}

Theorem: $K$ simplicial complex and $X \sqcup T \sqcup S$ a Morse matching.

There exists a chain complex $(𝒞(X), \partial^X)$ whose homology is the one of $K$:

\[∀ d, \qquad H_d(K) \, ≅ \, H_d(X)\]

NB: the dimensions are the same as in $K$, the homology groups remain the freely generated vector spaces of dimension $d$.

Def: Let $K = X \sqcup T \sqcup S$, let $τ, σ ∈ X$ be critical simplices st

\[\dim σ = \dim τ + 1\]

A gradient path from $σ$ to $τ$:

is a directed $σ$-$τ$-path in $\overline{H}$ (zigzaging between two consecutive dimensions) (cf. picture)

Def: Let $[σ : τ] ∈ ℤ/2ℤ$ be the number of distinct gradients paths from $σ$ to $τ$ $\mod 2$.

NB: $[σ : τ] = $

Finally, we set

\[\partial_p^X σ = \sum\limits_{ τ ∈ X, \dim τ = p-1 } [σ:τ] × t ∈ C_{p-1} (X) \\ \text{ where } \dim σ = p\]

NB: with $K = X$, we recover the orginal boundary maps, since there’s only one path from $σ$ to $τ$ (the edge $σ - τ$)

Prop: Let $K = X \sqcup T \sqcup S$.

\[∀ p, \qquad \partial_p^X \circ \partial_{p+1}^X = 0\]

Proof: Induction on $card \; T = card \; S$

Base case: $X = K; S, T = ∅$: $\partial$ of a simplicial complex ⟶ ok

Induction case: suppose we have a Morse matching.

\[K = X \sqcup T \sqcup S\]

and $\partial^X \circ \partial^X = 0$

Let $a$ be critical, $a ∈ X$.

\[\partial_d^X \partial_{d+1}^X a = \partial_d^X \Bigg(\sum\limits_{ b ∈ X_d } [a:b] b \Bigg) = \sum\limits_{ b ∈ X_d } \sum\limits_{ c ∈ X_{d-1} } [a:b] [b:c] c = 0\]

In particular: fix $a, c$ with

• $\dim a = d+1$
• $\dim c = d-1$

then

\[\sum\limits_{ b ∈ X_d } [a:b] [b:c] = 0\\ ⟹ \sum\limits_{ b ∈ X_d, b ≠ τ } [a:b] [b:c] = [a:τ][τ:c]\]

(the $c$’s are basis elements)

Now, take $(σ, τ) ∈ X$ st

\[(X \backslash \lbrace σ, τ\rbrace) \sqcup (T \backslash \lbrace τ\rbrace) \sqcup (S \backslash \lbrace σ\rbrace)\]

is a Morse matching

$\dim σ = \dim τ + 1 = d+1$

Then:

\[\partial_d^{X'} \partial_{d+1}^{X'} a = \partial_d^{X'} \Bigg( \sum\limits_{ b ∈ X_d, b ≠ τ } [a:b]b + [a:τ][σ:b]b \Bigg) = \partial_d^{X'} \Bigg( \sum\limits_{ b ∈ X_d, b ≠ τ } ([a:b] + [a:τ][σ:b])b \Bigg) \\ = \sum\limits_{ b ∈ X_d, b ≠ τ } \Bigg( ([a:b] + [a:τ][σ:b]) \sum\limits_{ c ∈ X_{d-1} } [b:c]c \Bigg) \\ =\sum\limits_{ b ∈ X_d, b ≠ τ } \Bigg( ([a:b] + [a:τ][σ:b]) \sum\limits_{ c ∈ X_{d-1} } [b:c]c \Bigg) \\ = \sum\limits_{ c ∈ X_{d-1} } \underbrace{\Bigg( \sum\limits_{ b ∈ X_d, b ≠ τ } [a:b][b:c] \Bigg)}_{= \, [a:τ][τ:c]} c + [a:τ] \sum\limits_{ c ∈ X_{d-1} } \underbrace{\Bigg( \sum\limits_{ b ∈ X_d, b ≠ τ } [σ:b][b:c] \Bigg)}_{= \, [σ:τ][τ:c]} c\\ = \sum\limits_{ c ∈ X_{d-1} } ([a:τ][τ:c] + [a:τ] \underbrace{[σ:τ]}_{\rlap{= 1 \text{ because otherwise, when flipping around the edge } (σ, τ), \text{ we get a cycle}}}[τ:c])c\\ = 0\]

Coming back to

Prop:

\[∀ d, \qquad H_d(K) \, ≅ \, H_d(X) \qquad K = X \sqcup T \sqcup S\]

Chain maps

A chain map between two chain complexes:

\[ϕ: (𝒞, \partial) ⟶ (𝒞', \partial')\]

is a collection of linear maps:

\[\lbrace ϕ_d : 𝒞_d ⟶ 𝒞'_d \rbrace_{d ∈ ℕ}\]

that commute with the boundary maps, i.e.

\[\begin{xy} \xymatrix{ 𝒞_d \ar[r]^{\partial} \ar[d]_{ϕ_d} & 𝒞_{d-1} \ar[d]^{ϕ_{d-1}} \\ 𝒞'_d \ar[r]_{\partial'} & 𝒞_{d-1} } \end{xy}\]

commutes

If $ϕ, ψ: (𝒞, \partial) ⟶ (𝒞’, \partial’)$:

a chain homotopy of $ϕ$ to $ψ$:

is a collection of linear maps

\[D_d: 𝒞_d ⟶ 𝒞_{d+1}'\]

such that

\[\partial'_{d+1} \circ D_p + D_{p-1} \circ \partial_d = ψ_d - ϕ_d\]

Diagrammatically:

\[\begin{xy} \xymatrix{ 𝒞_{d+1} \ar[r] & 𝒞_d \ar[ld]_{D_d} \ar[r]^{\partial_d} \ar[d]_{ϕ_d}^{ψ_d} & 𝒞_{d-1} \ar[ld]^{D_{d-1}} \\ 𝒞'_{d+1} \ar[r]_{\partial'_{d+1}} & 𝒞_d \ar[r] & 𝒞_{d-1} } \end{xy}\]

A chain map $ϕ: (𝒞, \partial) ⟶ (𝒞’, \partial’)$ is a chain equivalence:

if there exists a chain map $ϕ’: (𝒞’, \partial’) ⟶ (𝒞, \partial)$ such that $ϕ \circ ϕ’$ and $ϕ \circ ϕ$ are chain homotopic to the identity (resp. $id_{𝒞’}, id_𝒞$).

Thm: If there exists a chain equivalence between two complexes, they have the same homology.

Exercise: prove that an elementary collapse of a free pair $(τ, σ)$ preserves homology, using the following chain equivalence:

\[ϕ: 𝒞(K) ⟶ 𝒞(K \backslash \lbrace τ, σ\rbrace)\\ ψ: 𝒞(K \backslash \lbrace τ, σ\rbrace) ⟶ 𝒞(K) \qquad \text{ (standard inclusion)}\] \[ψ(x) = x \\ ϕ(x) = \begin{cases} 0 &&\text{ if } x = σ \\ x &&\text{ if } x ≠ τ, σ\\ \sum\limits_{ i } ξ_i &&\text{ if } x = τ \text{ st } \partial σ = τ + \sum\limits_{ i } ξ_i \end{cases}\]

using which $D_p$?

Then, do it for a Morse pair.