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The universal metric--trajectory projection is Fredholm

Statement

Fix distinct critical points $p, q$ of a Morse function on a closed manifold and take a sufficiently large finite $C^{h}$ Banach manifold $G^{h}$ of metrics fixed near the critical points. The zero set of the universal gradient-flow section near parametrized connecting trajectories is a Banach manifold, and its projection to $G^{h}$ is Fredholm of index $λ (p) - λ (q)$ . The free time-translation quotient is a Banach manifold, and its induced projection is Fredholm of index $λ (p) - λ (q) - 1$ .

Facts & Assumptions

Given: The finite- $C^{h}$ metric completion and the universal decaying trajectory section.

[F1]

The fixed-metric linearized operator detects stable--unstable transversality (Morse--Smale transversality and surjectivity of the linearized flow operator).

[F2]

Fredholm maps and their indices have the stated Banach-manifold meaning (Fredholm maps and regular values on countable-base Banach manifolds).

Proof

technique · direct

1.1

Metric variations supported where $d f \neq 0$ supply the missing cokernel directions of the fixed-metric operator in [F1]; consequently the universal section linearization is onto.

F1given

2.1

The Banach implicit-function theorem therefore makes its zero set a Banach manifold. Eliminating the trajectory variable leaves a Fredholm projection to the metric parameter, in the sense of [F2].

F2step 1.1

3.1

The asymptotic Morse splitting computes the parametrized projection's index as $λ (p) - λ (q)$ . Time translation is free and contributes the one-dimensional kernel direction; passing to the quotient therefore lowers the induced projection's index to $λ (p) - λ (q) - 1$ .

step 2.1algebra∎

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Sources

Alberto Abbondandolo and Pietro Majer, Lectures on the Morse Complex, Lemmas 2.23--2.24 (standard reference, not scraped)