Introduction to Morse theory and Morse homology
In this work we present a study of Morse theory with the aim of introducing the Morse homology theorem as its natural extension. For this we prove the classic Morse theorem, which states that a Morse function defined on a manifold determines its topology as CW-complex through its critical points. Ne...
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| Formato: | tesis de maestría |
| Estado: | Versión publicada |
| Fecha de publicación: | 2021 |
| País: | Brasil |
| Recursos: | Universidade de São Paulo (USP) |
| Repositorio: | Biblioteca Digital de Teses e Dissertações da USP |
| Idioma: | inglés |
| OAI Identifier: | oai:teses.usp.br:tde-20122021-113618 |
| Acesso em linha: | https://www.teses.usp.br/teses/disponiveis/55/55135/tde-20122021-113618/ |
| Access Level: | acceso abierto |
| Palavra-chave: | Funções de Morse-Smale Homologia de Morse Morse homology Morse theory Morse-Smale functions Teoria de Morse Topologia da variedade Topology of manifold |
| Resumo: | In this work we present a study of Morse theory with the aim of introducing the Morse homology theorem as its natural extension. For this we prove the classic Morse theorem, which states that a Morse function defined on a manifold determines its topology as CW-complex through its critical points. Next, we introduce the Morse and Poincaré polynomials, their relations, and the perfect Morse functions that show when the number of non-degenerate critical points is equal to the k-th Betti number of the manifold. Finally, we present the stable and unstable manifolds given by the gradient flow of a Morse-Smale function and the Morse-Smale-Witten chain complex whose homology is isomorphic to the singular homology of the manifold. |
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