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Twisted Morse complexes

Augustin Banyaga, David Hurtubise, Peter Spaeth

TL;DR

Twisted Morse complexes extend Morse homology and cohomology to local coefficient systems on closed manifolds, defining twisted Morse–Smale–Witten chains and cochains. The authors prove invariance and establish a Morse-theoretic Eilenberg theorem, showing twisted Morse homology coincides with singular (and Steenrod CW) homology with local coefficients, while twisted Morse cohomology matches Lichnerowicz cohomology for closed 1-forms. The framework unifies several domains, including regular CW-structures, sheaf cohomology, H-space obstructions, and Novikov theory, and yields concrete computations (e.g., Lichnerowicz cohomology on surfaces) and general obstructions to H-space structures. This approach provides practical tools for computing local-coefficient (co)homology via finite chain/cochain complexes and clarifies the conformal invariance of LCS-related invariants through eta-twisted theories.

Abstract

In this paper we study Morse homology and cohomology with local coefficients, i.e. "twisted" Morse homology and cohomology, on closed finite dimensional smooth manifolds. We prove a Morse theoretic version of Eilenberg's Theorem, and we prove isomorphisms between twisted Morse homology, Steenrod's CW-homology with local coefficients for regular CW-complexes, and singular homology with local coefficients. By proving Morse theoretic versions of the Poincare Lemma and of the de Rham Theorem, we show that twisted Morse cohomology with coefficients in a local system determined by a closed 1-form is isomorphic to the Lichnerowicz cohomology obtained by deforming the de Rham differential by the 1-form. We demonstrate the effectiveness of twisted Morse complexes by using them to compute Lichnerowicz cohomology, to compute obstructions to spaces being associative H-spaces, and to compute Novikov numbers.

Twisted Morse complexes

TL;DR

Twisted Morse complexes extend Morse homology and cohomology to local coefficient systems on closed manifolds, defining twisted Morse–Smale–Witten chains and cochains. The authors prove invariance and establish a Morse-theoretic Eilenberg theorem, showing twisted Morse homology coincides with singular (and Steenrod CW) homology with local coefficients, while twisted Morse cohomology matches Lichnerowicz cohomology for closed 1-forms. The framework unifies several domains, including regular CW-structures, sheaf cohomology, H-space obstructions, and Novikov theory, and yields concrete computations (e.g., Lichnerowicz cohomology on surfaces) and general obstructions to H-space structures. This approach provides practical tools for computing local-coefficient (co)homology via finite chain/cochain complexes and clarifies the conformal invariance of LCS-related invariants through eta-twisted theories.

Abstract

In this paper we study Morse homology and cohomology with local coefficients, i.e. "twisted" Morse homology and cohomology, on closed finite dimensional smooth manifolds. We prove a Morse theoretic version of Eilenberg's Theorem, and we prove isomorphisms between twisted Morse homology, Steenrod's CW-homology with local coefficients for regular CW-complexes, and singular homology with local coefficients. By proving Morse theoretic versions of the Poincare Lemma and of the de Rham Theorem, we show that twisted Morse cohomology with coefficients in a local system determined by a closed 1-form is isomorphic to the Lichnerowicz cohomology obtained by deforming the de Rham differential by the 1-form. We demonstrate the effectiveness of twisted Morse complexes by using them to compute Lichnerowicz cohomology, to compute obstructions to spaces being associative H-spaces, and to compute Novikov numbers.

Paper Structure

This paper contains 36 sections, 56 theorems, 314 equations, 23 figures.

Table of Contents

  1. Introduction
  2. The Morse complex with twisted coefficients
  3. Local coefficients
  4. Path components of compactified moduli spaces
  5. Twisting the Morse-Smale-Witten boundary operator
  6. Examples
  7. Computation of $H_0((C_\ast(f;G),\partial^G_\ast))$
  8. Morse Eilenberg Theorem
  9. The Morse-Smale-Witten chain complex on a covering space
  10. The Morse Eilenberg Theorem
  11. Homology determined by the isomorphism class of $G$
  12. A chain map
  13. A chain homotopy
  14. An invariance theorem
  15. Singular and CW-Homology with local coefficients
  16. ...and 21 more sections

Key Result

Theorem 1.2

Let $f:M \rightarrow \mathbb{R}$ be a smooth Morse-Smale function on a closed finite dimensional smooth Riemannian manifold $(M,\mathsf{g})$. Then the homology of the Morse-Smale-Witten chain complex with coefficients in $\mathbb{Z}$ is isomorphic to the singular homology of $M$ with coefficients in

Figures (23)

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  • ...and 18 more figures

Theorems & Definitions (122)

  • Definition 1.1: Morse-Smale-Witten chain complex
  • Theorem 1.2: Morse Homology Theorem
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Example 2.5: The local coefficient system $e^\eta$ determined by a closed $1$-form
  • Claim 2.6
  • Remark 2.7
  • Remark 2.8
  • ...and 112 more