Rational homotopy theory

• Ano de defesa: 2025
• Autor(a) principal: Oliveira, Bianca Carvalho de
• Tipo de documento: Dissertação
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Biblioteca Digitais de Teses e Dissertações da USP
• Palavras-chave em Português: Álgebras diferenciais graduadas e comutativas; Commutative differentially graded algebras; Conjuntos simpliciais; Rational homotopy theory; Simplicial sets; Teoria de homotopia
• Link de acesso: https://www.teses.usp.br/teses/disponiveis/45/45131/tde-23062025-163248/

The main objective of rational homotopy theory is to understand topological spaces up to torsion, and in this masters thesis we aim to create a concise yet comprehensive overview of this theory. To achieve this, we begin by introducing a foundational tool of the subject, namely the key concepts and results of abstract homotopy theory. Afterwards, we proceed to examine what particularities and obstacles rational homotopy theory offers, and how it should guide the methods we use. Then we will give a detailed construction of two model structures: on the category of simplicial sets, , and on the category of commutative differentially graded algebras over , . Having those two model structures, we show that they are Quillen equivalent, and so we can use one to obtain information about the other. Finally, we use the structure we obtained to build simple algebraic models to the rational homotopy types of simply connected spaces.