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 Rebiun34740314 https://catalogo.rebiun.org/rebiun/record/Rebiun34740314 cr ||||||a|a|| 240109s2023 sp o u spa DX1102328718 UEMC (21108) Xebook glg ES-AcoU spa Mosquera Lois, David Morse Theory on Finite Spaces David Mosquera Lois 1ª edición Santiago de Compostela Universidade de Santiago de Compostela. Servizo de Publicacións e Intercambio Científico 2023 Santiago de Compostela Santiago de Compostela Universidade de Santiago de Compostela. Servizo de Publicacións e Intercambio Científico 1 recurso electrónico 1 recurso electrónico Publicaciones del Departamento de Geometría y Topología Descrición do recurso : 9 de xaneiro de 2024 Universidade da Coruña 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 Modo de acceso: WWW Matemáticas LeBUC Publicaciones del Departamento de Geometría y Topología