Homotopical Cancellation Theory for Gutierrez-Sotomayor Singular Flows

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

Journal of Singularities, Tome 23 (2021), pp. 33-91

Résumé

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.

Détail

DOI : 10.5427/jsing.2021.23d

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