Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)In this article, Conley's connection matrix theory and a spectral sequence analysis of a filtered Morse chain complex (C, Delta) are used to study global continuation results for flows on surfaces. The briefly described unfoldings of Lyapunov graphs have been proved to be a well-suited combinatorial tool to keep track of continuations. The novelty herein is a global dynamical cancellation theorem inferred from the differentials of the spectral sequence (E-r, d(r)). The local version of this theorem relates differentials dr of the r th page E-r to Smale's theorem on cancellation of critical points.3617951838Fundacao de Amparo a Pesquisa do Estado de Sao Paulo (FAPESP) [2010/08579-0]Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPq) [302592/2010-5]FAPESP [2012/18780-0]Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)