Full text Author(s): Dahisy Valadão de Souza Lima Total Authors: 1 Document type: Doctoral Thesis Press: Campinas, SP. Institution: Universidade Estadual de Campinas (UNICAMP). Instituto de Matemática, Estatística e Computação Científica Defense date: 2014-07-23 Examining board members: Ketty Abaroa de Rezende; Marco Antonio Teixeira; Oziride Manzoli Neto; Regilene Delazari dos Santos Oliveira; Maria do Carmo Carbinatto Advisor: Ketty Abaroa de Rezende Abstract The main theme in this thesis is the study of gradient flows associated to a vector field $-\nabla f$ on closed manifolds, where $f$ is either a Morse function, a circle-valued Morse function or a Morse-Bott function. In order to obtain dynamical information, we make use of algebraic and topological tools such as spectral sequences and connection matrices. In the Morse context, consider a chain complex $(C,\Delta)$ generated by the critical points of $f$, where $\Delta$ counts the number of flow lines between consecutive critical points with signs. A spectral sequence $(E^{r},d^{r})$ analysis is used to obtain results on global continuation of flows on surfaces. A link is established between the differentials on the $r$-th page of $(E^{r},d^{r})$ and cancellation of critical points. In the circle-valued Morse case $f:M \rightarrow S^{1}$, a sweeping algorithm for the Novikov chain complex $(\mathcal{N},\Delta)$ associated to $f$ and generated by the critical points of $f$ is defined over the ring $\mathbb{Z}((t))$. This algorithm produces at each stage Novikov matrices. We prove that the last Novikov matrix has polynomial entries which is quite surprising since the matrices in the intermediary stages may have infinite series entries. We also present results showing that the modules and differentials of the spectral sequence associated to $(\mathcal{N},\Delta)$ can be retrieved through the sweeping algorithm. For gradient flows associated to Morse-Bott functions, the singularities form critical manifolds. We use the Conley index theory for the critical manifolds in order to characterize the set of connection matrices for Morse-Bott flows. Results are obtained on the effects on the set of connection matrices caused by a change in the partial ordering and Morse decomposition of isolated invariant sets (AU) FAPESP's process: 10/08579-0 - Transition Matrices associated with the Morse-Witten Complex Grantee: Dahisy Valadão de Souza Lima Support Opportunities: Scholarships in Brazil - Doctorate