Homotopical Cancellation Theory for Gutierrez-Sotomayor Singular Flows

In this article, we present a dynamical homotopical cancellation theory for Gutierrez-Sotomayor singular flows $\varphi$, GS-flows, on surfaces $M$. This generalizes the classical of Morse complexes smooth systems together with corresponding non-degenerate singularities. is accomplished by defining GS-chain complex $(M,\varphi)$ 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 within family $\{M_r,\varphi_r\}$ GS-flows $\varphi_r $ $M_r$, all which have same homotopy type as surprising element results that GS-singularities $\varphi_r$ are consonance associated sequence. Also, convergence corresponds GS-flow $\varphi_{\bar{r}}$ $M_{\bar{r}}$, some $\bar{r}$, property admits no further GS-singularities.