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The Morse complex is chain isomorphic to the handle cellular complex

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $(f, X)$ be Morse--Smale on a closed manifold $M$ , let ${e_{p}}_{p \in Crit (f)}$ be the cells of the Morse--Smale CW decomposition with the orientation-line orientations of the unstable manifolds (Compactified unstable manifolds give the Morse--Smale CW decomposition, The orientation line of a Morse critical point), and let $C_{*}^{c e l l} (M; Λ)$ be the cellular chain complex (Oriented cellular chain group, Cellular boundary from three consecutive skeleta, Cellular boundary is the incidence degree matrix, Cellular homology).

Then the sign-normalized identity map $Θ : C M_{k} (f, X; Λ) ⟶ C_{k}^{c e l l} (M; Λ), p ⟼ ε_{p}^{'} e_{p},$ with the signs $ε_{p}^{'} \in {\pm 1}$ of Cellular boundary coefficients are the signed trajectory counts, is an isomorphism of chain complexes over $Λ = Z / 2$ (where $ε_{p}^{'} = 1$ ) and over $Λ = Z$ . Equivalently, the signed trajectory counts agree, up to the index normalization, with the attaching-map (incidence) coefficients of the handle cellular complex; hence the Morse complex is chain isomorphic to the handle cellular complex, in particular chain homotopy equivalent to it (A chain isomorphism is a chain homotopy equivalence), and $H M_{k} (f, X; Λ) ≅ H_{k}^{c e l l} (M; Λ) .$

Facts & Assumptions

Given: The Axiom of Choice, Morse--Smale data $(f, X)$ on a closed manifold $M$ , the Morse complex and the cellular complex of its Morse--Smale CW decomposition.

[F1]

Both chain groups are free modules on one generator per critical point: the Morse chain group has basis ${Crit}_{k} (f)$ (The mod-two Morse chain group, The signed Morse differential over the integers) and the cellular chain group has basis the $k$ -cells ${e_{p}}$ , so $Θ$ is a degreewise isomorphism, with $ε_{p}^{'} \in {\pm 1}$ a diagonal change of basis in the integral case (Oriented cellular chain group, Cellular boundary from three consecutive skeleta, Chain complex in an abelian category).

[F2]

The Morse differential is the matrix of trajectory counts $n_{X} (p, q)$ over $Z$ and its mod-two reduction over $Z / 2$ (The signed Morse differential over the integers, The mod-two Morse differential); the cellular differential is the incidence matrix $[e_{p} : e_{q}]$ (Cellular boundary is the incidence degree matrix, Cellular boundary from three consecutive skeleta).

[F3]

The coefficient comparison of Cellular boundary coefficients are the signed trajectory counts gives $[e_{p} : e_{q}] = ε_{ind (p)} n_{X} (p, q)$ over $Z$ and $[e_{p} : e_{q}] \equiv # M (p, q)$ over $Z / 2$ , and states that the cellular boundary matrix equals the trajectory matrix after the diagonal basis change $e_{p} \mapsto ε_{p}^{'} e_{p}$ (Compactified unstable manifolds give the Morse--Smale CW decomposition for the CW structure used).

[F4]

A chain isomorphism is a chain homotopy equivalence, and isomorphisms of chain complexes induce isomorphisms on homology (A chain isomorphism is a chain homotopy equivalence, Chain complex in an abelian category, A graded morphism of chain complexes).

[F5]

The homology of the Morse complex is Morse homology and the homology of the cellular complex is cellular homology (Morse homology of a Morse--Smale pair, Cellular homology).

Proof

technique · direct

1.1F1given

By [F1] the map $Θ$ is a degreewise isomorphism of graded modules: on each degree it sends the basis ${Crit}_{k} (f)$ bijectively onto the basis of $k$ -cells, up to the diagonal signs $ε_{p}^{'}$ which are invertible in both coefficient rings.

2.1F2F3step 1.1

It remains to check that $Θ$ commutes with the differentials. By [F2] the two differentials are the trajectory matrix and the incidence matrix, and by [F3] the incidence matrix equals the trajectory matrix after the diagonal basis change $e_{p} \mapsto ε_{p}^{'} e_{p}$ in the integral case, while over $Z / 2$ the incidence coefficients are exactly the mod-two trajectory counts.

3.1F2F3step 2.1algebra

Therefore, for every $k$ , the composite $Θ \circ \partial_{k}^{M o r s e}$ and $\partial_{k}^{c e l l} \circ Θ$ have the same matrix with respect to the chosen bases; equality of matrices on a basis gives equality of homomorphisms, so $Θ$ is a morphism of chain complexes, and being degreewise invertible it is an isomorphism of chain complexes.

4.1F4F5step 3.1∎

By [F4] the chain isomorphism $Θ$ is a chain homotopy equivalence and induces an isomorphism on homology; by [F5] this homology isomorphism is the displayed $H M_{k} (f, X; Λ) ≅ H_{k}^{c e l l} (M; Λ)$ . This completes the proof.

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Sources

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