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Canonical chain complexes for Morse-Smale vector fields

Clemens Bannwart, Claudia Landi

TL;DR

The paper constructs a canonical chain complex $C_\bullet(v)$ associated to any Morse-Smale vector field $v$ on a closed manifold $M$ by turning the Čech-homology spectral sequence of Smale's unstable-manifolds filtration into a chain complex via exact-couple derived structures. This complex satisfies $H_*(C_\bullet(v)) \cong H_*(M)$ and has a basis determined by the fixed points and closed orbits of $v$, with the number of generators in degree $k$ equal to $| ext{Fix}_k(v)|+| ext{Orb}_{k-1}(v)|+| ext{Orb}_k(v)|$; it is invariant under topological equivalence and canonically defined without arbitrary choices. In the gradient-like case, $C_\bullet(v)$ recovers the classical Morse complex and thus the standard Morse homology, while the generalized Morse inequalities follow from the algebraic Morse inequalities applied to $C_\bullet(v)$. The paper also provides detailed constructions, proofs, and explicit 2-sphere examples to illustrate the method and demonstrates how the canonical generators on the first page arise from the dynamics.

Abstract

In 1960, Smale defined a filtration of a closed smooth manifold by the unstable manifolds of fixed points and closed orbits of a Morse-Smale vector field defined on it, and derived generalized Morse inequalities. This suggests that, similarly to the Morse chain complex of a gradient-like vector field, even in the presence of closed orbits, Morse-Smale vector fields admit canonical chain complexes, invariant under topological equivalence, from which one can algebraically derive Morse inequalities. In this paper we show that this is actually the case, improving the state of the art that only offers non-canonical chain complexes. Technically, we achieve this result considering the Čech homology spectral sequence of the unstable manifolds filtration. In particular, we turn bounded exact couples into chain complexes such that the limit page of the spectral sequence associated with an exact couple gives the homology of the chain complex. We showcase our construction with examples.

Canonical chain complexes for Morse-Smale vector fields

TL;DR

The paper constructs a canonical chain complex associated to any Morse-Smale vector field on a closed manifold by turning the Čech-homology spectral sequence of Smale's unstable-manifolds filtration into a chain complex via exact-couple derived structures. This complex satisfies and has a basis determined by the fixed points and closed orbits of , with the number of generators in degree equal to ; it is invariant under topological equivalence and canonically defined without arbitrary choices. In the gradient-like case, recovers the classical Morse complex and thus the standard Morse homology, while the generalized Morse inequalities follow from the algebraic Morse inequalities applied to . The paper also provides detailed constructions, proofs, and explicit 2-sphere examples to illustrate the method and demonstrates how the canonical generators on the first page arise from the dynamics.

Abstract

In 1960, Smale defined a filtration of a closed smooth manifold by the unstable manifolds of fixed points and closed orbits of a Morse-Smale vector field defined on it, and derived generalized Morse inequalities. This suggests that, similarly to the Morse chain complex of a gradient-like vector field, even in the presence of closed orbits, Morse-Smale vector fields admit canonical chain complexes, invariant under topological equivalence, from which one can algebraically derive Morse inequalities. In this paper we show that this is actually the case, improving the state of the art that only offers non-canonical chain complexes. Technically, we achieve this result considering the Čech homology spectral sequence of the unstable manifolds filtration. In particular, we turn bounded exact couples into chain complexes such that the limit page of the spectral sequence associated with an exact couple gives the homology of the chain complex. We showcase our construction with examples.
Paper Structure (19 sections, 37 theorems, 45 equations, 2 figures)

This paper contains 19 sections, 37 theorems, 45 equations, 2 figures.

Table of Contents

  1. The setting
  2. Stable and unstable manifolds
  3. Morse-Smale vector fields
  4. The unstable manifolds filtration of a Morse-Smale vector field
  5. Turning spectral sequences into chain complexes
  6. Preliminaries on spectral sequences and exact couples
  7. The chain complex of a spectral sequence (via exact couples)
  8. The spectral sequence of the unstable manifolds filtration
  9. Preliminaries on Čech homology
  10. The spectral sequence in Čech homology of a topological filtration
  11. Preparatory homology computations
  12. Canonical generators for the first page
  13. The chain complex of a Morse-Smale vector field
  14. Properties of $C_\bullet(v)$
  15. The gradient-like case
  16. ...and 4 more sections

Key Result

Proposition 1.2

Let $p\in M$. Then $\alpha_v(p)$ and $\omega_v(p)$ are non-empty, closed, connected and $\phi_v$-invariant subsets of $M$.

Figures (2)

  • Figure 1: Morse-Smale vector field on $S^2$ with three sinks, one saddle, and one repelling orbit
  • Figure 2: Morse-Smale vector field on $S^2$ with four sinks, four saddles, two sources, and one repelling orbit

Theorems & Definitions (75)

  • Definition 1.1
  • Proposition 1.2: Proposition 1.4 of palis2012geometric
  • Definition 1.3
  • Remark 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.7
  • Lemma 1.8: Lemma 3.8 in Smale1960MorseIineq
  • Definition 1.9
  • Remark 1.10
  • ...and 65 more