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Gluing continuation solutions gives collar neighbourhoods of the broken ends

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $(f_{s}, g_{s})$ be a regular continuation datum on a closed manifold $M$ , let $ind (p) - ind (q) = 1$ , and let $β = (γ^{-}, v) \in M^{-} (p, a) \times C (a, q), ind (a) = ind (q),$ or $β = (v, γ^{+}) \in C (p, b) \times M^{+} (b, q), ind (b) = ind (p),$ be a once-broken continuation trajectory (Broken continuation trajectories and geometric convergence). Then there are $δ > 0$ and a continuous injection $ψ : [0, δ) \to \overline{C} (p, q)$ with $ψ (0) = β, ψ (t) \in C (p, q) (t > 0),$ that is smooth on $(0, δ)$ , whose parameter $t$ is a neck-length (gluing) parameter, and whose image is a neighbourhood of $β$ ; moreover every sequence in $C (p, q)$ converging geometrically to $β$ lies eventually in $ψ ((0, δ))$ . Consequently every once-broken boundary point of the compactification of Continuation trajectories are compact up to breaking has a one-sided collar chart, and $\overline{C} (p, q)$ is a compact one-dimensional topological manifold with boundary whose boundary is exactly the disjoint union of these once-broken products.

Facts & Assumptions

Given: The Axiom of Choice, the stated regular datum and index-drop-one once-broken configuration.

[F1]

Its geometric compactification is compact metrizable, and the only added configurations are the two displayed once-broken products. The rigid middle and metric-end tail sets are finite (Continuation trajectories are compact up to breaking, Broken continuation trajectories and geometric convergence).

[F2]

The local exact flow-matching chart has one coordinate $ρ = 1 / T$ at a single break, with a fixed-time anchor adjacent to the unshifted middle. It includes constant connectors. The chart is a homeomorphism onto a geometric neighbourhood and every sufficiently close trajectory has the unique inverse crossing-time coordinate (Finite flow matching gives local charts at metric-end broken trajectories).

[F3]

The regular unbroken space has dimension one (A regular continuation datum between Morse--Smale pairs).

Proof

technique · direct

1.1F1F2givenconstruct

At the specified once-broken point, the two rigid piece spaces have singleton local charts by [F1]. The broken-stratum chart $B$ of [F2] is therefore a point. Its one-neck matching chart is a homeomorphism $ψ : [0, δ) \to U$ , taking zero to the broken point and positive $ρ$ to exact unbroken solutions. It is smooth on the positive interval. No middle translation is used: the passage ends, or starts in the positive-break case, at the fixed-time anchor. This remains valid if the middle is a constant solution at the intermediate critical point.

2.1F2step 1.1

By the inverse coverage of [F2], any ordinary trajectory geometrically sufficiently close to the broken point has the unique entry-crossing time relative to that anchor, hence the unique $T$ and $ρ = 1 / T$ . It is the matched trajectory $ψ (ρ)$ . Thus the image is a neighbourhood, the map is injective, and every sequence converging to that broken point lies eventually in its positive image.

3.1F1F3step 1.1step 2.1∎

Apply steps 1.1–2.1 at each added point. By [F1] these are exactly the once-broken products, and every such point has a half-interval neighbourhood. The unbroken points have ordinary one-dimensional charts by [F3]. Together with compactness and metrizability of [F1], these charts make the compactification a compact one-dimensional topological manifold with boundary in the sense of Topological manifolds with boundary, with the stated boundary and collars.

Depends on

Used by

Dependency tree · two levels

48 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

Udhav Fowdar, A Functional Analytic Approach to Morse Homology (UCL 4th-year project, complete PDF, 93 pp.) (standard reference, not scraped)
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)