On the classifying space of a Morse flow category

TL;DR

This paper proves that the classifying space of a tame Morse flow category accurately recovers the homotopy type of a smooth, closed manifold, clarifying the importance of tameness assumptions in Morse theory.

Contribution

It establishes that the flow category's classifying space recovers the manifold's homotopy type under tameness conditions and demonstrates failure without these assumptions.

Findings

Classifying space recovers homotopy type for tame Morse functions

Tameness assumption is essential for correct homotopy recovery

Counterexample on S^2×S^1 shows failure without tameness

Abstract

We show that the classifying space of the flow category of a \emph{tame} Morse function on a smooth, closed manifold $M$ recovers the homotopy type of $M$, thereby addressing a claim in a preprint of Cohen--Jones--Segal. The tameness assumption is that the compactified moduli spaces of broken gradient trajectories are locally contractible, ensuring the flow category is topologically well-behaved. We construct a Morse function and Riemannian metric on $S^2\times S^1$ for which the associated flow category fails to recover the correct homotopy type, showing that the tameness hypothesis is crucial. Together, these results clarify the extent to which transversality assumptions can be relaxed so that the flow category models the homotopy type of the underlying manifold.