Transition Matrices associated with the Morse-Witten Complex

Abstract

In this project we will adopt a Morse-Witten approach via Conley index theory to associate the topology of the manifold to the trajectories connecting critical points of a gradient flow which define the moduli spaces. Some of these flow connections are registered in the integer entry connection matrix. We will study the connection and transition matrices which appear in the Sweeping Algorithm applied to this differential, the initial connection matrix, with the objective of better understanding the dynamical behaviour at each step. In particular, we would like to understand the bifurcations which occur in the transition matrix stages as well as investigate the effects on the compactified moduli spaces of higher dimensions.