Singularity Collisions through Homotopical Dynamical cancellation

TL;DR

This paper develops a homotopical framework for analyzing singular dynamical systems, extending classical Morse cancellation theory to GGS flows on GGS manifolds, with algebraic and spectral sequence tools.

Contribution

It introduces a GGS chain complex and spectral sequence analysis to connect algebraic cancellations with homotopical dynamical cancellations in singular flows.

Findings

Established a bijective correspondence between spectral sequence cancellations and dynamical cancellations.

Extended Morse cancellation theory to singular GGS flows.

Provided illustrative examples demonstrating the framework's applicability.

Abstract

We introduce collisions of invariant sets and, in particular, consider dynamical homotopical cancellations that preserve the homotopy type of the underlying singular manifold. We develop the theory of homotopical dynamical cancellation for generalized Gutierrez-Sotomayor (GGS) flows defined on GGS manifolds. This framework extends the classical cancellation theory of Morse flows to the singular setting. To effectively capture these homotopical cancellations, we introduce a GGS chain complex, which encodes essential dynamical and algebraic-topological information. Furthermore, we provide a spectral sequence analysis of a filtered GGS chain complex, demonstrating a bijective correspondence between algebraic cancellations of the modules of the spectral sequence and homotopical dynamical cancellations in the GGS flow. Several illustrative examples are presented, highlighting the practical applicability of the proposed framework.