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On the classifying space of a Morse flow category

Maxine E. Calle, Fangji Liu

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.

On the classifying space of a Morse flow category

Abstract

We show that the classifying space of the flow category of a \emph{tame} Morse function on a smooth, closed manifold recovers the homotopy type of , 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 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.
Paper Structure (14 sections, 24 theorems, 98 equations, 13 figures)

This paper contains 14 sections, 24 theorems, 98 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Preliminaries on Morse theory
  4. Morse theory and gradient flow
  5. The compactified moduli space of flows
  6. The Cohen-Jones-Segal Theorem
  7. The flow category
  8. Some intermediary constructions
  9. Proof of \ref{['thm:loc']}
  10. A counterexample to the general claim
  11. A perturbation on the Morse flows
  12. A counterexample
  13. Background on classifying spaces of topological categories
  14. Topologies on compactified moduli spaces of gradient flows

Key Result

Theorem 1.1

Let $(M,g)$ be a smooth, closed, finite-dimensional Riemannian manifold. If $f\colon M\to \mathbb{R}$ is a tame Morse function, then there is a homotopy equivalence $B\mathscr{C}_f\simeq M$.

Figures (13)

  • Figure 1: \ref{['assum']}
  • Figure 2: The box $N\times[-1,1]\times[0,1]\subseteq M$
  • Figure 3: Perturbation of the vector field at $x=x_0$
  • Figure 4: The perturbed moduli space
  • Figure 5: $S^2\times S^1\subset \mathbb{R}^4$
  • ...and 8 more figures

Theorems & Definitions (79)

  • Theorem 1.1: \ref{['thm:loc']}
  • Theorem 1.2: \ref{['thm:counter']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • proof
  • ...and 69 more