Morse homology for the Hamiltonian action in cotangent bundles - Publication - Bridge of Knowledge

Search

Morse homology for the Hamiltonian action in cotangent bundles

Abstract

In this paper we use the gradient flow equation introduced in Asselle and Starostka (Calc. Var. 59: 113, 2020) to construct a Morse complex for the Hamiltonian action AH 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 −∇AH of the Hamiltonian action within a class of pseudogradient vector fields preserving all good compactness properties of −∇AH . 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.

Citations

  • 0

    CrossRef

  • 0

    Web of Science

  • 0

    Scopus

Cite as

Full text

full text is not available in portal

Keywords

Details

Category:
Articles
Type:
artykuły w czasopismach
Published in:
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS no. 64,
ISSN: 0944-2669
Language:
English
Publication year:
2025
Bibliographic description:
Asselle L., Starostka M.: Morse homology for the Hamiltonian action in cotangent bundles// CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS -,iss. 4 (2025),
DOI:
Digital Object Identifier (open in new tab) 10.1007/s00526-025-02953-3
Sources of funding:
  • Free publication
Verified by:
Gdańsk University of Technology

seen 0 times

Recommended for you

Meta Tags