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Morse homology recovers the Morse inequalities

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $M$ be a closed manifold, let $(f, X)$ be Morse--Smale on $M$ , write $c_{k} = # {Crit}_{k} (f)$ (Nondegenerate critical points, nullity, index, and coindex, A Morse function on a compact manifold has finitely many critical points) and put $b_{k} = \dim_{Λ} H_{k} (M; Λ)$ for a field $Λ$ (Morse homology is naturally isomorphic to singular homology, Cellular homology). Then:

$c_{k} \geq b_{k}$ for every $k$ ;
$\sum_{i = 0}^{k} (- 1)^{k - i} c_{i} \geq \sum_{i = 0}^{k} (- 1)^{k - i} b_{i}$ for every $k$ ;
with $c (t) = \sum_{k} c_{k} t^{k}$ , $b (t) = \sum_{k} b_{k} t^{k}$ and $r_{k} = {rank}_{Λ} \partial_{k}$ the rank of the Morse differential, $c (t) = b (t) + (1 + t) Q (t)$ where $Q (t) = \sum_{k} r_{k + 1} t^{k}$ has nonnegative integer coefficients (The mod-two Morse chain group, The signed Morse differential over the integers, Morse homology of a Morse--Smale pair);
$\sum_{k} (- 1)^{k} c_{k} = χ (M) = \sum_{k} (- 1)^{k} b_{k}$ (Euler characteristic of a finite CW complex, Euler–Poincare formula for finite CW complexes, Compactified unstable manifolds give the Morse--Smale CW decomposition).

Over $Z$ the same identities hold with $b_{k}$ replaced by the rank of the finitely generated group $H_{k} (M; Z)$ .

Facts & Assumptions

Given: The Axiom of Choice, a closed manifold $M$ , a Morse--Smale pair $(f, X)$ , and a field $Λ$ (or $Λ = Z$ ).

[F1]

For a field $Λ$ , take the finite free complex on the critical points whose differential is the integral signed trajectory matrix with its entries mapped from $Z$ to $Λ$ . The coefficient comparison identifies it, after the index sign normalization, with the cellular complex with constant coefficients $Λ$ (Cellular boundary coefficients are the signed trajectory counts). The constant-local-system cellular theorem computes $H_{*} (M; Λ)$ (Cellular chains compute local homology, Homology and cohomology with local coefficients). Thus its chain ranks are $c_{k}$ , its homology dimensions are $b_{k}$ , and its differential ranks $r_{k}$ are nonnegative integers. Over $Z$ the same identification uses the integral Morse complex (The signed Morse differential over the integers, Morse homology of a Morse--Smale pair) and identifies its homology with $H_{*} (M; Z)$ .

[F2]

Rank-nullity for a finite-dimensional linear map gives $\dim \ker \partial_{k} = c_{k} - r_{k}$ ; the homology in degree $k$ is $\ker \partial_{k} / im \partial_{k + 1}$ , so $b_{k} = c_{k} - r_{k} - r_{k + 1}$ ; this pure linear algebra uses only the finite dimensions supplied by [F1].

[F3]

The Morse--Smale CW decomposition has exactly $c_{k}$ cells in dimension $k$ , and the Euler--Poincar'e formula identifies the alternating sum of the cell counts with the Euler characteristic and with the alternating sum of the homology ranks (Compactified unstable manifolds give the Morse--Smale CW decomposition, Euler characteristic of a finite CW complex, Euler–Poincare formula for finite CW complexes).

Verification

technique · direct

1.1F1F2givenalgebra

By [F1] the numbers $c_{k}$ , $b_{k}$ and $r_{k}$ are well-defined nonnegative integers; by rank-nullity as recorded in [F2], $c_{k} - r_{k} = \dim \ker \partial_{k} = b_{k} + r_{k + 1}$ , hence $c_{k} = b_{k} + r_{k} + r_{k + 1}$ for every $k$ , which immediately gives $c_{k} \geq b_{k}$ ; this is (1).

2.1step 1.1algebra

Multiplying $c_{i} = b_{i} + r_{i} + r_{i + 1}$ by $(- 1)^{k - i}$ and summing over $0 \leq i \leq k$ , the coefficients of $r_{1}, \dots, r_{k}$ cancel in pairs and only the $r_{0}$ - and $r_{k + 1}$ -terms survive, giving $\sum_{i = 0}^{k} (- 1)^{k - i} c_{i} = \sum_{i = 0}^{k} (- 1)^{k - i} b_{i} + (- 1)^{k} r_{0} + r_{k + 1}$ ; since $r_{0} = 0$ and $r_{k + 1} \geq 0$ , the last term is nonnegative, which gives (2).

2.2step 1.1algebra

Substituting $c_{k} = b_{k} + r_{k} + r_{k + 1}$ into the generating series and using $\sum_{k} r_{k} t^{k} = t Q (t)$ and $\sum_{k} r_{k + 1} t^{k} = Q (t)$ gives the polynomial identity $c (t) = b (t) + (1 + t) Q (t)$ with $Q (t) = \sum_{k} r_{k + 1} t^{k} \geq 0$ coefficientwise; this is (3).

2.3F3step 1.1

By [F3] the alternating sum $\sum_{k} (- 1)^{k} c_{k}$ is the Euler characteristic of the CW complex with $c_{k}$ cells in dimension $k$ , hence equals $χ (M)$ , and the Euler--Poincar'e formula also gives $χ (M) = \sum_{k} (- 1)^{k} b_{k}$ ; this is (4).

3.1F1step 2.2∎

Over $Z$ the chain groups are free of finite rank and rank-nullity for finitely generated abelian groups gives the same identities with $b_{k}$ the rank of $H_{k} (M; Z)$ ; the identification $rank H M_{k} = rank H_{k} (M; Z)$ is the comparison theorem of [F1], so (1)--(4) hold verbatim.

Depends on

Used by

Continuation across a birth--death pair adds an acyclic summand Example

Dependency tree · two levels

108 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Alexander F. Ritter, Part III Morse Homology (Cambridge lecture notes, complete author PDF, 115 pp.) (standard reference, not scraped)
Liviu I. Nicolaescu, An Invitation to Morse Theory, 2nd ed. (complete author PDF, 291 pp.) (standard reference, not scraped)
Michele Audin and Mihai Damian, Morse Theory and Floer Homology (complete author PDF of the English book, 628 pp.) (standard reference, not scraped)