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Morse homology for the Hamiltonian action in cotangent bundles

L. Asselle, M. Starostka

TL;DR

The paper develops a Morse-theoretic framework for the Hamiltonian action $\mathbb{A}_H$ on a mixed-regularity loop space in cotangent bundles $T^*M$, overcoming infinite Morse indices by leveraging a strongly integrable (0)-essential subbundle $\mathcal{E}^s$. A genuine Morse complex is built from finite-dimensional intersections of unstable and stable manifolds, with transversality achieved through generic perturbations of a negative pseudo-gradient that preserve essential compactness. The resulting Morse homology is independent of the Hamiltonian and is canonically isomorphic to both Floer homology of $T^*M$ and the singular homology of the free loop space $LM$, providing a robust, geometry-based alternative to Floer theory. The approach offers potential generalizations to broader symplectic settings and emphasizes transversality and precompactness as central advantages of the Morse-theoretic route in settings where Floer theory faces technical obstructions.

Abstract

In this paper we use the gradient flow equation introduced in [10] to construct a Morse complex for the Hamiltonian action $\mathbb A_H$ on a mixed regularity space of loops in the cotangent bundle $T^*M$ of a closed manifold $M$. Connections between pairs of critical points are realized as genuine intersections between unstable and stable manifolds, which (despite being infinite dimensional objects) turn out to have finite dimensional intersection with good compactness properties. This follows from the existence of an additional structure, namely a strongly integrable (0)-essential subbundle, which behaves nicely under the negative gradient flow of the Hamiltonian action and which is needed to make comparisons. Transversality is achieved by generically perturbing the negative gradient vector field $-\nabla \mathbb A_H$ of the Hamiltonian action within a class of pseudo-gradient vector fields preserving all good compactness properties of $-\nabla \mathbb A_H$. This follows from an abstract transversality result of independent interest for vector fields on a Hilbert manifold for which stable and unstable manifolds of rest points are infinite dimensional. The resulting Morse homology is independent of the choice of the Hamiltonian (and of all other choices but the choice of the (0)-essential subbundle, which however only changes the Morse-complex by a shift of the indices) and is isomorphic to the Floer homology of $T^*M$ as well as to the singular homology of the free loop space of $M$.

Morse homology for the Hamiltonian action in cotangent bundles

TL;DR

The paper develops a Morse-theoretic framework for the Hamiltonian action on a mixed-regularity loop space in cotangent bundles , overcoming infinite Morse indices by leveraging a strongly integrable (0)-essential subbundle . A genuine Morse complex is built from finite-dimensional intersections of unstable and stable manifolds, with transversality achieved through generic perturbations of a negative pseudo-gradient that preserve essential compactness. The resulting Morse homology is independent of the Hamiltonian and is canonically isomorphic to both Floer homology of and the singular homology of the free loop space , providing a robust, geometry-based alternative to Floer theory. The approach offers potential generalizations to broader symplectic settings and emphasizes transversality and precompactness as central advantages of the Morse-theoretic route in settings where Floer theory faces technical obstructions.

Abstract

In this paper we use the gradient flow equation introduced in [10] to construct a Morse complex for the Hamiltonian action on a mixed regularity space of loops in the cotangent bundle of a closed manifold . Connections between pairs of critical points are realized as genuine intersections between unstable and stable manifolds, which (despite being infinite dimensional objects) turn out to have finite dimensional intersection with good compactness properties. This follows from the existence of an additional structure, namely a strongly integrable (0)-essential subbundle, which behaves nicely under the negative gradient flow of the Hamiltonian action and which is needed to make comparisons. Transversality is achieved by generically perturbing the negative gradient vector field of the Hamiltonian action within a class of pseudo-gradient vector fields preserving all good compactness properties of . This follows from an abstract transversality result of independent interest for vector fields on a Hilbert manifold for which stable and unstable manifolds of rest points are infinite dimensional. The resulting Morse homology is independent of the choice of the Hamiltonian (and of all other choices but the choice of the (0)-essential subbundle, which however only changes the Morse-complex by a shift of the indices) and is isomorphic to the Floer homology of as well as to the singular homology of the free loop space of .
Paper Structure (12 sections, 18 theorems, 139 equations)

This paper contains 12 sections, 18 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The functional setting
  4. Stable and unstable manifolds
  5. Essential subbundles
  6. Some preliminary computations
  7. The (0)-essential subbundle $\mathcal{E}$
  8. Pre-compactness of intersection between stable and unstable manifold
  9. The space of vector fields that preserve an essentially vertical family
  10. Proof of precompactness
  11. Transversality
  12. General facts about transversality

Key Result

Theorem 1.2

Let $M$ be a closed manifold, and let $H:\mathds{T}\times T^*M \to \mathds{R}$ be a smooth Hamiltonian which is fiberwise convex and quadratic outside a compact set, see eq:growthcondition. Then, for every $s\in (1/2,3/4)$, there is a well-defined Morse complex with $\mathds{Z}_2$-coefficients for t

Theorems & Definitions (38)

  • Remark 1.1
  • Theorem 1.2
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 28 more