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The continuation count is a chain map

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 $M$ , let $Φ$ be its continuation map (The continuation chain map) and let $\partial^{\pm}$ be the Morse differentials over $Λ = Z / 2$ and $Λ = Z$ (The mod-two Morse differential, The signed Morse differential over the integers).

Then $Φ$ is a chain map: $Φ_{k - 1} \circ \partial_{k}^{-} = \partial_{k}^{+} \circ Φ_{k} for every k,$ in both coefficient cases, with any chosen positive rays in the critical orientation lines in the integral case. Consequently $Φ$ induces a homomorphism $[Φ] : H M_{k} (f^{-}, g^{-}; Λ) \to H M_{k} (f^{+}, g^{+}; Λ)$ on Morse homology (Morse homology of a Morse--Smale pair, Chain complex in an abelian category, A graded morphism of chain complexes).

Facts & Assumptions

Given: The Axiom of Choice, a regular continuation datum $(f_{s}, g_{s})$ with continuation map $Φ$ , and the two Morse differentials.

[F1]

For critical points with index drop one the compactification is a compact one-manifold with boundary, whose boundary is the disjoint union of the once-broken products $M^{-} (p, a) \times C (a, q)$ and $C (p, b) \times M^{+} (b, q)$ with the prescribed index conditions, and each end moduli space occurring is finite (Continuation trajectories are compact up to breaking, Gluing continuation solutions gives collar neighbourhoods of the broken ends).

[F2]

A compact one-manifold has an even number of boundary points (Boundary of a compact 1-manifold has even cardinality), and a compact oriented one-manifold has signed boundary count zero (Oriented boundary counts of a compact oriented 1-manifold cancel).

[F3]

With the orientation conventions of Orientation lines orient the continuation moduli spaces compatibly with gluing, the sign of a once-broken boundary point of the type $(γ^{-}, v)$ is $τ (γ^{-}) τ (v)$ and the sign of one of the type $(v, γ^{+})$ is $- τ (v) τ (γ^{+})$ , the relative sign being the ordered fixed-anchor endpoint comparison; the differential conventions of The signed Morse differential over the integers are then such that the signed boundary sum of the boundary points of $\overline{C} (p, q)$ is the coefficient of $q$ in $(Φ_{k - 1} \circ \partial_{k}^{-} - \partial_{k}^{+} \circ Φ_{k}) (p)$ .

[F4]

The differentials are the trajectory counts of The mod-two Morse differential and The signed Morse differential over the integers; over $Z / 2$ all signs are absent, and the continuation map is the count of The continuation chain map; this dictionary identifies the two sides of the coefficient computations below. Both maps are graded: $\partial^{\pm}$ and $Φ$ lower, respectively preserve, the index, so coefficients vanish automatically outside the two cases considered.

[F5]

The actual metric-end differentials are defined and square to zero with these count and sign conventions (Arbitrary metric Morse--Smale end counts form finite Morse chain complexes).

Proof

technique · direct

1.1F1F5given

Fix $k$ , a generator $p \in {Crit}_{k} (f^{-})$ and a generator $q \in {Crit}_{k - 1} (f^{+})$ , so that $ind (p) - ind (q) = 1$ . By [F1] the compactification $\overline{C} (p, q)$ is a compact one-manifold with boundary, and its boundary is the disjoint union of the once-broken products $M^{-} (p, a) \times C (a, q)$ and $C (p, b) \times M^{+} (b, q)$ , each of them finite.

2.1F2F4step 1.1

Over $Z / 2$ , the boundary of $\overline{C} (p, q)$ has an even number of points by [F2]; counting the boundary points by their type gives $\sum_{a} # M^{-} (p, a) # C (a, q) + \sum_{b} # C (p, b) # M^{+} (b, q) \equiv 0$ , which by the dictionary of [F4] is the $q$ -coefficient of $Φ_{k - 1} \partial^{-} p + \partial_{k}^{+} Φ p$ ; over $Z / 2$ this sum vanishes, which is the asserted identity in the mod-two case.

2.2F2F3step 1.1

Over $Z$ , [F2] makes the signed boundary count of the oriented compactification vanish; by [F3] the boundary signs are the products of the piece signs, and the differential and orientation normalization identifies this signed count with the $q$ -coefficient of $(Φ_{k - 1} \partial^{-} - \partial_{k}^{+} Φ) (p)$ , which therefore vanishes.

3.1F4step 2.1step 2.2

Steps 2.1 and 2.2 cover every pair $(p, q)$ of generators with $ind (p) = ind (q) + 1$ , and for all other pairs both sides of the identity have zero coefficient by the grading recorded in [F4]; extending linearly over the generators gives $Φ_{k - 1} \circ \partial_{k}^{-} = \partial_{k}^{+} \circ Φ_{k}$ for every $k$ over both coefficient rings, i.e. a degree-zero morphism of chain complexes.

4.1step 3.1∎

A morphism of chain complexes induces a homomorphism on homology by the universal property of the homology functor, so $[Φ] : H M_{k} (f^{-}, g^{-}; Λ) \to H M_{k} (f^{+}, g^{+}; Λ)$ is well defined; this is the last assertion.

Depends on

Used by

Dependency tree · two levels

70 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)
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)