On an article by S. A. Barannikov

TL;DR

This paper introduces the Morse-Barannikov complex, a simplified canonical Morse complex that makes homology computation straightforward, and explores its bifurcation behavior in families of functions with applications to boundary manifolds.

Contribution

It presents the Morse-Barannikov complex as a canonical, simplified Morse complex and studies its bifurcation theory in parameterized families of functions.

Findings

The Morse-Barannikov complex provides an immediate reading of homology.

Bifurcation analysis of the complex in one-parameter families is developed.

Applications to boundary manifolds demonstrate practical relevance.

Abstract

Given a Morse function f on a closed manifold M with distinct critical values, and given a field F, there is a canonical complex, called the Morse-Barannikov complex, which is equivalent to any Morse complex associated with f and whose form is simple. In particular, the homology of M with coefficients in F is immediately readable on this complex. The bifurcation theory of this complex in a generic one-parameter family of functions will be investigated. Applications to the boundary manifolds will be given.