  In this article, we present a dynamical homotopical cancellation theory for
Gutierrez-Sotomayor singular flows $\varphi$, 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 $(M,\varphi)$ 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 algebraic cancellations
within the spectral sequence to a family $\{M_r,\varphi_r\}$ of GS-flows
$\varphi_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 $\varphi_r$ 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 $\varphi_{\bar{r}}$ on $M_{\bar{r}}$, for some
$\bar{r}$, with the property that $\varphi_{\bar{r}}$ admits no further
dynamical homotopical cancellation of GS-singularities.
