Topological Complexity and the Lusternik-Schnirelmann Category

• Autor(a) principal: Lier, Matias de Jong van
• Data de Publicação: 2021
• Tipo de documento: Dissertação
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: https://www.teses.usp.br/teses/disponiveis/55/55135/tde-09092021-104209/
• topic: Algebraic topology; Categoria de Lusternik-Schnirelmann; Complexidade topológica; Fibrewise topology; Genus de Schwarz; Lusternik-Schnirelmann category; Motion planning problem; Problema do planejamento de movimento; Schwarz genus; Topologia algébrica; Topologia fibracional; Topological complexity

In recent years, a new field integrating robotics and topology was born, referred to by many as Topological Robotics, in which the main strategy is to use algebraic topological tools to get some insight into robotics problems. One of those problems is called the robot motion planning problem and is the main motivation for this work. We present an in-depth study of Topological Complexity, discussing how it relates to the Motion Planning Problem, and the main methods for computing it for CW complexes and Smooth Manifolds, spaces of great interest in robotics. The concept of Lusternik- Schnirelmann category is introduced due to its connection with Topological complexity, both being particular cases of the more general concept of the Schwarz Genus of a fibration.