Descripción del título
Morse Theory on Finite Spac...
The topology of a space and the kind of dynamics that can be defined on it are strongly interconnected. Precisely, both Morse theory and Lusternik-Schnirelmann theory arose to study this phenomenon. Accordingly, we devote the first part of this thesis to the study of Morse theory and Lusternik-Schnirelmann theory in some discrete settings - finite topological spaces, partially ordered sets, chain complexes, cell complexes, and certain small categories. We show the relation between different existing Morse theories in these settings, and we develop them in more depth. Among the results, we prove a set of Morse inequalities for non-necessarily gradient dynamics, involving torsion in the coefficients. Besides, we obtain a Lusternik-Schnirelmann theorem in this context. In the second part of the thesis, we introduce a new notion that we have called "homotopic distance", which generalizes classical homotopic invariants such as the Lusternik-Schnirelmann category and Topological Complexity. Moreover, it provides deeper relations between them
Monografía
monografia Rebiun36700156 https://catalogo.rebiun.org/rebiun/record/Rebiun36700156 cr ||||||a|a|| 231002s230206 |||||||||||00| 0 spa d DX1102328718 spa Morse Theory on Finite Spaces David Mosquera Lois 1 Santiago de Compostela Universidade de Santiago de Compostela. Servizo de Publicacións e Intercambio Científico 2023-02-06 Santiago de Compostela Santiago de Compostela Universidade de Santiago de Compostela. Servizo de Publicacións e Intercambio Científico Publicaciones del Departamento de Geometría y Topología The topology of a space and the kind of dynamics that can be defined on it are strongly interconnected. Precisely, both Morse theory and Lusternik-Schnirelmann theory arose to study this phenomenon. Accordingly, we devote the first part of this thesis to the study of Morse theory and Lusternik-Schnirelmann theory in some discrete settings - finite topological spaces, partially ordered sets, chain complexes, cell complexes, and certain small categories. We show the relation between different existing Morse theories in these settings, and we develop them in more depth. Among the results, we prove a set of Morse inequalities for non-necessarily gradient dynamics, involving torsion in the coefficients. Besides, we obtain a Lusternik-Schnirelmann theorem in this context. In the second part of the thesis, we introduce a new notion that we have called "homotopic distance", which generalizes classical homotopic invariants such as the Lusternik-Schnirelmann category and Topological Complexity. Moreover, it provides deeper relations between them Morse Theory Mathematics Matemáticas Mathematics Matemáticas Mosquera Lois, David