HOMOTOPICAL CANCELLATION THEORY FOR GUTIERREZ-SOTOMAYOR SINGULAR FLOWS

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((r) over bar) on M-(r) over bar, for some (r) over bar, with the property that phi((r) over bar) admits no further dynamical homotopical cancellation of GS-singularities. (AU)