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Reverse continuation is an inverse on Morse homology

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $(f_{s}, g_{s})$ be a regular continuation datum from $(f^{-}, g^{-})$ to $(f^{+}, g^{+})$ on a closed manifold $M$ , and let $({\overset{ˉ}{f}}_{s}, {\overset{ˉ}{g}}_{s})$ be a regular continuation datum from $(f^{+}, g^{+})$ to $(f^{-}, g^{-})$ obtained from the reversed family $(f_{- s}, g_{- s})$ by a sufficiently small generic perturbation fixing the two ends (regular data are residual in the space of paths with fixed ends, as recorded in the definition); denote the continuation maps by $Φ$ and $\overset{ˉ}{Φ}$ (The continuation chain map, A regular continuation datum between Morse--Smale pairs).

Then $[\overset{ˉ}{Φ}] \circ [Φ] = {id}_{H M_{*} (f^{-}, g^{-}; Λ)}, [Φ] \circ [\overset{ˉ}{Φ}] = {id}_{H M_{*} (f^{+}, g^{+}; Λ)},$ so $[Φ] : H M_{*} (f^{-}, g^{-}; Λ) \to H M_{*} (f^{+}, g^{+}; Λ)$ is an isomorphism with inverse $[\overset{ˉ}{Φ}]$ (Morse homology of a Morse--Smale pair). In particular the Morse homologies of any two Morse--Smale pairs on a closed manifold are canonically isomorphic by continuation.

Facts & Assumptions

Given: The Axiom of Choice, a regular continuation datum $(f_{s}, g_{s})$ from $(f^{-}, g^{-})$ to $(f^{+}, g^{+})$ , and a regular continuation datum $({\overset{ˉ}{f}}_{s}, {\overset{ˉ}{g}}_{s})$ from $(f^{+}, g^{+})$ to $(f^{-}, g^{-})$ obtained by a small generic perturbation, fixing the ends, of the reversed family $(f_{- s}, g_{- s})$ .

[F1]

The reversed family $({\overset{ˉ}{f}}_{s}^{(0)}, {\overset{ˉ}{g}}_{s}^{(0)}) : = (f_{- s}, g_{- s})$ is a continuation datum from $(f^{+}, g^{+})$ to $(f^{-}, g^{-})$ : it is smooth, it equals $(f^{+}, g^{+})$ for $s \leq - S$ and $(f^{-}, g^{-})$ for $s \geq S$ , and its continuation equation is $\partial_{s} v = - \nabla^{{\overset{ˉ}{g}}_{s}^{(0)}} {\overset{ˉ}{f}}_{s}^{(0)} (v)$ . Reversing the parameter in a solution $u$ of the original equation gives $\frac{d}{d s} (u (- s)) = + \nabla^{g_{- s}} f_{- s} (u (- s))$ , the positive-gradient equation, so the solutions of the reversed family are not the time reversals of the solutions of the original equation and regularity of the reversed family is a separate condition. When both the function path and the metric path may vary, regular data form a residual set with fixed ends. Thus the reversed family can be perturbed arbitrarily little in both components, fixing its two ends, to a regular datum; the datum $({\overset{ˉ}{f}}_{s}, {\overset{ˉ}{g}}_{s})$ of the statement is such a perturbation (A regular continuation datum between Morse--Smale pairs).

[F2]

The composition law: the composite of the continuation maps along a spliced datum is the homology map of that spliced datum, which is independent of the splicing choices (Composition of continuation maps on homology, A regular two-parameter continuation datum).

[F3]

The spliced datum for the pair of reverse data from $(f^{-}, g^{-})$ back to $(f^{-}, g^{-})$ is joined by a regular two-parameter family to the constant datum, so the two continuation maps are chain homotopic; the constant datum's continuation map is the identity (Homotopic continuation data give chain homotopic maps, The continuation map of constant data is the identity).

[F4]

Chain homotopic maps induce the same map on homology, and the identity on a chain complex induces the identity on homology (Morse homology of a Morse--Smale pair, The continuation chain map).

Proof

technique · direct

1.1F1given

By [F1] $({\overset{ˉ}{f}}_{s}, {\overset{ˉ}{g}}_{s})$ is a regular continuation datum from $(f^{+}, g^{+})$ to $(f^{-}, g^{-})$ , so both $Φ$ and $\overset{ˉ}{Φ}$ are well-defined continuation maps and induce maps on Morse homology; the argument below uses only that the two data join the same two end pairs, in opposite directions.

2.1F2step 1.1

Apply the composition law of [F2] to the pair $(Φ, \overset{ˉ}{Φ})$ in the order from $(f^{-}, g^{-})$ to $(f^{+}, g^{+})$ and back: there is a regular spliced datum from $(f^{-}, g^{-})$ to itself whose homology map equals $[\overset{ˉ}{Φ}] \circ [Φ]$ .

3.1F3step 2.1

Applying [F3] to that spliced datum connects it by a regular two-parameter family to the constant datum; hence $[\overset{ˉ}{Φ}] \circ [Φ]$ equals the homology map of the constant datum, which is the identity. This gives the first identity.

4.1F4step 3.1∎

The same argument with the roles of the two pairs exchanged gives $[Φ] \circ [\overset{ˉ}{Φ}] = {id}_{H M_{*} (f^{+}, g^{+}; Λ)}$ ; the two identities together say that $[Φ]$ is an isomorphism with inverse $[\overset{ˉ}{Φ}]$ , which is the last assertion.

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Sources

Alexander F. Ritter, Part III Morse Homology (Cambridge lecture notes, complete author PDF, 115 pp.) (standard reference, not scraped)
Michael Hutchings, Math 242 Lecture 21: Invariance via continuation maps (notes by Jackson Van Dyke, complete PDF) (standard reference, not scraped)
Udhav Fowdar, A Functional Analytic Approach to Morse Homology (UCL 4th-year project, complete PDF, 93 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)