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A parametrized Morse trajectory space is a manifold

Statement

Let $(f, X)$ be Morse--Smale on an $n$ -manifold. For distinct critical points $p, q$ , $\tilde{M} (p, q)$ is a smooth manifold of dimension $λ (p) - λ (q)$ (and is empty if the transverse fibre product is empty).

Facts & Assumptions

Given: A Morse--Smale pair $(f, X)$ and distinct critical points $p, q$ .

[F1]

The global unstable and stable manifolds are immersed of dimensions $λ (p)$ and $n - λ (q)$ (Global stable and unstable manifolds are immersed Euclidean spaces).

[F2]

A transverse fibre product of smooth maps is an embedded submanifold of the product (Transverse fibre products are embedded submanifolds).

Proof

technique · direct

1.1

The Morse--Smale condition says that the two immersion maps from $W^{u} (p)$ and $W^{s} (q)$ are transverse. Thus [F2] makes their fibre product a smooth manifold. Its evaluation image is precisely $W^{u} (p) \cap W^{s} (q)$ , hence, by the point-marked convention, $\tilde{M} (p, q)$ .

F2given

2.1

The dimension of that transverse fibre product is $λ (p) + (n - λ (q)) - n = λ (p) - λ (q)$ by [F1].

F1step 1.1algebra∎

Depends on

Used by

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Sources

Michèle Audin and Mihai Damian, Morse Theory and Floer Homology, §2.2.b (standard reference, not scraped)