Morse homology: orientation of the moduli space of gradient flow lines, coherence and applications

TL;DR

This paper develops a method to compute Morse homology over integers by carefully orienting moduli spaces of gradient flow lines and applies it to 4-manifolds, advancing the understanding of Morse theoretical invariants.

Contribution

It introduces a precise orientation construction for moduli spaces in Morse homology and applies it to compute homology groups of 4-manifolds.

Findings

Successfully computes Morse homology over integers for specific cases.

Provides a new orientation framework for moduli spaces in Morse theory.

Demonstrates applications to 4-manifold topology.

Abstract

In this paper, we shall compute the chain complex and the corresponding homology of some Morse function $f$ over integer coefficients. The definition of the correct boundary operator requires a careful construction of moduli space of (pseudo)gradient flow lines orientations. We will then apply this construction in the computation of these homology groups on 4-manifolds.