LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedaudited 2026-09-07

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Descending flow identifies the local and global attaching regions

Statement

Assume ${A C}_{ω}$ . Let $f^{- 1} ([a, b])$ be compact, with regular endpoints and exactly one critical point $p$ , nondegenerate of index $k$ , with value $c = f (p)$ . For the local Morse attaching embedding on $M_{c - ε}$ , where $a < c - ε < c$ , descending flow transports its entire thickening to $M_{a}$ as an embedded framed attaching region, provided there is no intervening critical value. The regular regions outside the local critical model are identified by collars.

Facts & Assumptions

[F1]

Local morse sublevel pair is a handle pair: Assume ${A C}_{ω}$ . In a sufficiently small Morse chart $f = c - ∣ u ∣^{2} + ∣ v ∣^{2}$ , with $u \in R^{k}$ and $v \in R^{n - k}$ , the change across $c$ is a rounded index- $k$ handle: a compact product piece attaches along $S^{k - 1} \times D^{n - k}$ on $f = c - ε$ , its core is $v = 0$ , and after a local modification the remaining region up to $c + ε$ is a regular collar. The modification agrees with $f$ off a compact subset of the chart. The collar assertion is made inside a compact band having no other critical point.

[F2]

Adapted descending field near a compact morse band: Assume ${A C}_{ω}$ . Suppose the compact closed band of a smooth function on a boundaryless manifold has only finitely many critical points, all nondegenerate. There is a smooth field $X$ with $d f (X) < 0$ at every noncritical point of the band and $X = (2 u, - 2 v)$ in smaller disjoint Morse charts $f = f (p) - ∣ u ∣^{2} + ∣ v ∣^{2}$ . It can be chosen compactly supported on $M$ and hence complete.

[F3]

Regular interval diffeomorphism: Assume ${A C}_{ω}$ . If $a < b$ and the closed band $K = f^{- 1} ([a, b])$ of a smooth function on a boundaryless manifold is compact and critical-point-free, its normalized flow gives a level-preserving diffeomorphism $T : M_{a} \times [a, b] \to K$ , $T (x, t) = Φ_{t - a} (x)$ .

[F4]

The fundamental theorem on flows: Let $X$ be a smooth vector field on $M$ . For each $p \in M$ , let $γ_{p} : I_{p} \to M$ be the maximal integral curve through $p$ , and set $D : = {(t, p) \in R \times M : t \in I_{p}}, Φ (t, p) : = γ_{p} (t) .$ Then $D$ is open in $R \times M$ , each fibre $D_{p}$ is an interval containing $0$ , the map $Φ : D \to M$ is smooth, and $Φ$ is the unique maximal local flow generated by $X$ .

Proof

Given: The objects and hypotheses in the statement.

1.1

Choose the local attaching tube and the adapted complete descending field $X$ . If the regular band $f^{- 1} ([a, c - ε])$ is empty, both levels and the attaching region are empty and transport is vacuous. Otherwise, on that compact band $- d f (X) > 0$ has a positive minimum. Normalize to $Z = X / (- d f (X))$ near that band and cut off outside a relatively compact regular neighborhood. Its flow $Ψ$ has $d f (Z) = - 1$ throughout the band; finite-time continuation follows exactly as for the regular product.

F1F2F3

2.1

Let $h$ be the local tube embedding. Define $h_{a} (z) = Ψ_{c - ε - a} (h (z))$ . Reverse flow for the same time is its inverse onto the image. Smooth dependence and uniqueness show that this is an embedding, including its disk boundary. The derivative sends the chosen normal disk coordinates to linearly independent normal coordinates, since the level-to-level map is a diffeomorphism. It therefore transports the framing, not just the sphere.

F4step 1.1

3.1

The same flow gives the collar between $M_{a}$ and $M_{c - ε}$ . Near the critical model its complementary collar is the regular modified-function collar from the local lemma; outside the compact modification support the modified function equals $f$ . On that common regular region choose its descending collar field to agree with $Z$ : patch $Z$ with any descending field for the modified function using a smooth partition, as in the adapted-field construction, and normalize by the negative derivative of the modified function. The convex combination remains descending. Choose the partition to retain $Z$ on a smaller neighborhood of the chart interface. Uniqueness of flow then makes the collar charts agree on their common flow neighborhoods there. If the attaching region is empty there is nothing to transport. These arguments use neither orientation nor Morse–Smale transversality.

F1F2F3F4step 2.1∎

Depends on

Used by

Unstable disk is the handle core Corollary
One critical point handle attachment Theorem

Dependency tree · two levels

18 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

Benedetti, Lectures on Differential Topology (standard reference, not scraped)