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

• Published in: Ergodic theory and dynamical systems Vol. 36; no. 6; pp. 1795 - 1838
• Main Authors: BERTOLIM, M. A., LIMA, D. V. S., MELLO, M. P., DE REZENDE, K. A., DA SILVEIRA, M. R.
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.09.2016
• Subjects: Analysis; Cancellation; Combinatorial analysis; Differentials; Dynamical systems; Matrix; Matrix theory; Spectra; Theorems; Theoretical mathematics; Topology; Tracking
• ISSN: 0143-3857; 1469-4417
• DOI: 10.1017/etds.2014.142

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.