CounterexampleConstruction: Literature-sourcedVerification: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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An index-one moduli locus can be infinite without compactness

Statement refuted

Without a compactness hypothesis, discrete index-one Morse trajectory data are finite.

Witness

Let $M = ∐_{n \geq 1} R_{n}$ and on $R_{n}$ put

$f_{n} (x) = \frac{n}{2} (3 x - x^{3}), p_{n} = 1, q_{n} = - 1.$

Choose the locally normalized bounded downward-gradient-like field described by $X_{n} = 2 u \partial_{u}$ near $p_{n}$ , $X_{n} = - 2 v \partial_{v}$ near $q_{n}$ , and $X_{n} = - f_{n}^{'} / (1 + (f_{n}^{'})^{2}) \partial_{x}$ away from those charts, patched by disjoint nonnegative bump functions. It is complete. Each component has one unparametrized trajectory from the index-one maximum $p_{n}$ to the index-zero minimum $q_{n}$ ; hence the global index-one locus is an infinite discrete union.

Facts & Assumptions

Given: The above disjoint-union field, with the local Morse coordinates and bump-function patching specified in the example.

[F1]

A downward gradient-like field has the stated exact local normal forms and strictly decreases $f$ off critical points (Downward gradient-like vector fields for a Morse function).

Counterexample

technique · direct

1.1

Near $p_{n}$ and $q_{n}$ the displayed local fields are the required Morse normal forms; off them every patched summand has $d f_{n} (X_{n}) < 0$ . The coefficients are bounded on fixed supports and the outside coefficient has absolute value at most $1 / 2$ , so every $X_{n}$ is complete.

F1given

2.1

On each line the interval $(- 1, 1)$ is one flow orbit from $p_{n}$ to $q_{n}$ . Thus $M (p_{n}, q_{n})$ is a singleton and hence is discrete, without any appeal to Morse--Smale transversality.

step 1.1

3.1

Therefore $M_{1} (X) : = ∐_{n \geq 1} M (p_{n}, q_{n})$ is an infinite discrete set. It refutes finiteness of the global index-one locus without a compactness condition, not finiteness for one fixed endpoint pair.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Kai Cieliebak and Urs Frauenfelder, Morse homology on noncompact manifolds, Introduction example (standard reference, not scraped)