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The index-two compactification is a compact one-manifold with boundary

Statement

Assume the Axiom of Choice. Let $(f, X)$ be Morse--Smale on a closed manifold $M$ and let $λ (p) - λ (q) = 2$ . Then $\overline{M} (p, q)$ , with the geometric-convergence topology, is a compact metrizable second-countable smooth $1$ -manifold with boundary whose interior is $M (p, q)$ , and $\partial \overline{M} (p, q) = ⨆_{r : λ (r) = λ (p) - 1} M (p, r) \times M (r, q),$ a finite disjoint union of finite discrete spaces. Every boundary point has the one-sided collar chart $[0, δ)$ of Gluing once-broken index-two trajectories: collar ends, so the compactification is obtained from the one-dimensional smooth manifold $M (p, q)$ by adding the once-broken trajectories.

Facts & Assumptions

Given: The Axiom of Choice, a Morse--Smale pair $(f, X)$ on a closed manifold $M$ , and critical points $p, q$ with $λ (p) - λ (q) = 2$ .

[A1]

The Axiom of Choice (The Axiom of Choice).

[F1]

$M (p, q)$ is a smooth manifold of dimension $λ (p) - λ (q) - 1 = 1$ (Index-two trajectory spaces are one-dimensional, The unparametrized trajectory space is a smooth manifold).

[F2]

$\overline{M} (p, q)$ is compact, metrizable and second-countable in the geometric-convergence topology, and $M (p, q)$ is open and dense in it (Compactness up to breaking of Morse trajectory spaces, Geometric convergence to a broken trajectory).

[F3]

In index drop two every broken trajectory of length at least two is once-broken with exactly one intermediate critical point, of index $λ (p) - 1$ ; broken trajectories of length one are the elements of $M (p, q)$ (Breaking length is bounded by the index drop, Broken Morse trajectories).

[F4]

For index drop one the moduli space is finite, so each factor $M (p, r)$ and $M (r, q)$ with $λ (r) = λ (p) - 1$ is finite, and only finitely many intermediate critical points $r$ occur (Index-one trajectory moduli spaces are finite, A Morse function on a compact manifold has finitely many critical points, Nondegenerate critical points, nullity, index, and coindex).

[F5]

Each once-broken trajectory with $λ (r) = λ (p) - 1$ has a one-sided collar chart $ψ : [0, δ) \to \overline{M} (p, q)$ with $ψ (0)$ the broken point, $ψ (s) \in M (p, q)$ for $s > 0$ , $ψ$ smooth on $(0, δ)$ and injective, whose image is a neighbourhood and which captures every geometrically convergent sequence (Gluing once-broken index-two trajectories: collar ends, [A1]).

[F6]

A topological $n$ -manifold with boundary is locally modelled on the half-space, and its boundary is the set of points whose charts have last coordinate $0$ ; a smooth structure with boundary is an atlas of such charts with smooth transitions (Topological manifolds with boundary, Smooth charts, atlases, and structures with boundary).

Proof

technique · direct

1.1F1F2

By [F1] the interior part $M (p, q)$ is a smooth $1$ -manifold, and by [F2] it is open in $\overline{M} (p, q)$ . Its points are interior points of the compactification: the charts of the manifold structure of [F1] are charts of $\overline{M} (p, q)$ around them.

2.1A1F3F5F6step 1.1

The added points are the broken trajectories of length at least two, and by [F3] each of them is a once-broken trajectory with intermediate point $r$ of index $λ (p) - 1$ . Conversely every once-broken trajectory with such an $r$ is a broken trajectory of length two and is not ordinary, hence an added point. So the boundary set is exactly $⨆_{λ (r) = λ (p) - 1} M (p, r) \times M (r, q)$ as a set, and each of its points has a collar chart $[0, δ)$ by [F5]. There are finitely many added points by [F4], and the metric topology of [F2] allows the collars to be shrunk to pairwise disjoint neighbourhoods. On an overlap with an interior chart, the collar and its inverse are smooth because its interior restriction is a smooth embedding of one-dimensional manifolds. There are no overlaps between different boundary charts after this shrinking. Thus the collars and interior charts give a compatible smooth atlas as required by [F6]; hence $\overline{M} (p, q)$ is a smooth $1$ -manifold with boundary whose interior is $M (p, q)$ and whose boundary is that set.

3.1F3F4step 2.1

The boundary identification is a homeomorphism onto the disjoint union of the products: the factors are discrete spaces (by the index-one finiteness and discreteness in [F4]), finitely many by [F4], and a sequence of once-broken trajectories with a fixed intermediate point $r$ converges geometrically to $(γ_{1}, γ_{2})$ exactly when its components converge to $γ_{1}$ and $γ_{2}$ in the geometric topologies, which is the product of the discrete topologies; distinct intermediate points $r$ give disjoint factors because a once-broken trajectory determines its intermediate critical point. Since each factor is a finite discrete space by [F4], the boundary is a finite disjoint union of finite discrete spaces.

4.1A1F2F5step 2.1step 3.1∎

Compactness, metrizability and second countability are [F2]; the collar charts of [F5] show that the compactification is obtained from $M (p, q)$ by adding the once-broken trajectories, completing the proof.

Depends on

Used by

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Sources

Michele Audin and Mihai Damian, Morse Theory and Floer Homology, Part I Ch. 3, complete author PDF (standard reference, not scraped)
Alexander F. Ritter, Part III Morse Homology (Cambridge lecture notes), Lectures 17-19, complete combined PDF (standard reference, not scraped)
Ralph L. Cohen, Bundles, Manifolds, and Homotopy, Ch. 13 and Appendix A, complete author PDF (standard reference, not scraped)