On handle theory for Morse–Bott critical manifolds

Abstract

In this article, we use Conley Index Theory to set up a framework to associate topological-dynamical invariants to a Morse–Bott flow in a neighborhood of a critical manifold. Isolating blocks N for a critical manifold S are characterized in terms of conditions on the ranks of its homology Conley index. The necessity of these conditions follows from the generalized Morse–Bott inequalities for isolating blocks. Morse–Bott semi-graphs turn out to be a useful combinatorial device to record these topological conditions. One goal is to verify that these conditions when imposed on an n-abstract Morse–Bott semi-graph are sufficient for its realization. This is attained by introducing Morse–Bott handle surgeries in order to construct isolating blocks for critical manifolds in dimensions 2 and 3, and for a large class in higher dimensions. Stronger results are obtained in dimensions 2 and 3 where necessary and sufficient conditions for a Morse–Bott graph to be associated to a Morse–Bott flow on some manifold M are determined.