Homotopical Cancellation Theory for Gutierrez-Sotomayor Singular Flows

Abstract

In this article, we present a dynamical homotopical cancellation theory for Gutierrez-Sotomayor singular flows $\phi$, 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 $(M,\phi )$ 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 $\{M_r,{\phi }_r\}$ of GS-flows ${\phi }_r$ 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 ${\phi }_r$ 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 ${\phi }_{\bar{r}}$ on $M_{\bar{r}}$, for some $\bar{r}$, with the property that ${\phi }_{\bar{r}}$ admits no further dynamical homotopical cancellation of GS-singularities.