Flow and compactness hypotheses for noncompact Morse homology

The closedness hypotheses in A regular continuation datum between Morse--Smale pairs, Continuation trajectories are compact up to breaking and Canonical Morse homology of a closed manifold are not formal. On a noncompact manifold:

1. a continuation trajectory can escape to spatial infinity along a direction in which the interpolation is nonconstant, so that no limit or broken continuation trajectory need exist, and the compactness-up-to-breaking theorem has no analogue without a properness or compactness package controlling the ends (Compactness up to breaking needs closedness or a proper compactness package);
2. a non-proper Morse function can have infinitely many critical points and unbounded trajectory moduli, so the chain groups need not be finitely generated and the coefficient sums defining the differentials and the continuation maps need not converge;
3. an incomplete metric or field lets trajectories reach infinity in finite time, and the two-end limits then fail; completeness of the flow is an extra hypothesis in the noncompact case (Completeness of a gradient flow is an extra hypothesis on a noncompact manifold);
4. the uniform energy bound of the continuation energy identity is useless without a compactness (Palais--Smale-type) condition on the relevant trajectory sets.

Consequently the Morse complex, the continuation maps and the gluing/compactification results require additional structure, such as properness or an exhaustion with compact Morse slabs (Proper smooth functions and compact Morse slabs, Proper Morse slabs prevent finite-time escape of connecting trajectories) and a compactness package controlling the broken ends. The counterexample of the companion page displays the escape mechanism concretely; the positive noncompact theory is the Floer/Morse theory of proper or exhaustion-controlled data and is not asserted here.

Properness is one way to obtain the required controls, not a necessary condition in every noncompact example. This item records the scope boundary of the closed theory: the individual failure mechanisms are those of the cited remarks and propositions, the companion counterexample exhibits the escape concretely, and no positive noncompact theorem is asserted here.