A global two-dimensional version of Smale’s cancellation theorem via spectral sequences

Abstract

In this article, Conley’s connection matrix theory and a spectral sequence analysis of a filtered Morse chain complex $(C,{\rm\Delta})$ are used to study global continuation results for flows on surfaces. The briefly described unfoldings of Lyapunov graphs have been proved to be a well-suited combinatorial tool to keep track of continuations. The novelty herein is a global dynamical cancellation theorem inferred from the differentials of the spectral sequence $(E^{r},d^{r})$. The local version of this theorem relates differentials $d^{r}$ of the $r$th page $E^{r}$ to Smale’s theorem on cancellation of critical points.