Homotopical Cancellation Theory for Gutierrez-Sotomayor Singular Flows

D.V.S. Lima, S. A. Raminelli, and K. A. de Rezende

Journal of Singularities
volume 23 (2021), 33-91

Received: 28 April 2020. Received in revised form: 5 January 2021.

DOI: 10.5427/jsing.2021.23d

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Abstract:

In this article, we present a dynamical homotopical cancellation theory for Gutierrez-Sotomayor singular flows φ, GS-flows, on singular surfaces M. This theory generalizes the classical theory of Morse complexes of smooth dynamical systems together with the corresponding cancellation theory for non-degenerate singularities. This is accomplished by defining a GS-chain complex for (M,φ) and computing its spectral sequence (E^r, d^r). As r increases, algebraic cancellations occur, causing modules in E^r to become trivial. The main theorems herein relate these algebraic cancellations within the spectral sequence to a family {M_r, φ_r} of GS-flows φ_r on singular surfaces M_r, all of which have the same homotopy type as M. The surprising element in these results is that the dynamical homotopical cancellation of GS-singularities of the flows φ_r are in consonance with the algebraic cancellation of the modules in E^r of its associated spectral sequence. Also, the convergence of the spectral sequence corresponds to a GS-flow φ_{\bar{r}} on M_{\bar{r}}, for some \bar{r}, with the property that φ_{\bar{r}} admits no further dynamical homotopical cancellation of GS-singularities.

2010 Mathematical Subject Classification:

58K45, 58k60, 58K65, 55U15, 55T05, 37B30, 37D15

Key words and phrases:

GS-singularities, stratified manifold, chain complexes, spectral sequence, Gutierrez-Sotomayor flows, dynamical homotopical cancellation

Author(s) information:

D.V.S. Lima
CMCC
Universidade Federal do ABC
Santo André, SP, Brazil
email: dahisy.lima@ufabc.edu.br

S. A. Raminelli
IMECC
Universidade Estadual de Campinas
Campinas, SP, Brazil
email: stefakemi@gmail.com

K. A. de Rezende
IMECC
Universidade Estadual de Campinas
Campinas, SP, Brazil
email: ketty@ime.unicamp.br