The Morse–Witten complex via dynamical systems

Abstract

Given a smooth closed manifold M, the Morse–Witten complex associated to a Morse function f and a Riemannian metric g on M consists of chain groups generated by the critical points of f and a boundary operator counting isolated flow lines of the negative gradient flow. Its homology reproduces singular homology of M. The geometric approach presented here was developed in Weber [Der Morse–Witten Komplex, Diploma Thesis, TU Berlin, 1993] and is based on tools from hyperbolic dynamical systems. For instance, we apply the Grobman–Hartman theorem and the λ -lemma (Inclination Lemma) to analyze compactness and define gluing for the moduli space of flow lines.