An introduction to rational homotopy theory

Abstract

Rational homotopy theory is the study of homotopy groups modulo torsion. The idea is to consider the torsion-free part of $\pi_n(X)$ by tensoring by $\mathbb{Q}$, so computations become much more affordable. The aim of this work is to provide a fairly detailed introduction to rational homotopy theory from Sullivan's approach. We will start by introducing the concept of rationalization from both the topological and algebraic points of view. Second, we will construct the functor piece-wise linear forms, which establish the link between topology and algebra associating to each simply connected space $X$ a commutative differential graded algebra $A_{\mathrm{pl}}(X)$. However, the key part of this theory is to associate to $A_{\mathrm{pl}}(X)$ a much more simple type of cdga's: Sullivan algebras, which allows us to do computations explicitly. Finally, given a fibration $F \hookrightarrow F \rightarrow B$ we will study the relation between the Sullivan models of each space.