Twisted Morse complexes

TL;DR

This paper develops a Morse theoretic framework for twisted Morse homology and cohomology with local coefficients, establishing isomorphisms with classical homology theories and applying these to compute various topological invariants.

Contribution

It introduces Morse theoretic versions of classical theorems for twisted homology, linking Morse complexes with Steenrod, singular, and Lichnerowicz cohomologies, and demonstrates their computational utility.

Findings

Proves Morse theoretic versions of Eilenberg's Theorem, Poincare Lemma, and de Rham Theorem.

Establishes isomorphisms between twisted Morse homology, Steenrod CW-homology, and singular homology.

Uses twisted Morse complexes to compute Lichnerowicz cohomology, obstructions to H-space structures, and Novikov numbers.

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.