Assar Andersson: Modelling rational homotopy types with $A_\infty$-algebras

Abstract:

We start by introducing the reader to $A_\infty$-algebras, and their morphisms. We show that $A_\infty$-algebras over a graded vector space $V$ corresponds to cofree DGA-coalgebras over $V$. Next, we use homological perturbation theory to prove the homotopy transfer theorem. 

We also give an outline of why, simply connected topological spaces $X$,$Y$ with finite dimensional $H^p(X;\mathbb{Q})$, $H^p(Y;\mathbb{Q})$, for each $p$, are of the same rational homotopy type if and only if $C^*(X;\mathbb{Q})$ and $C^*(Y;\mathbb{Q})$ are of the same homotopy type.

Finally, we use the homotopy transfer theorem to model cochain algebras with $A_\infty$-algebras over their homology, and we show that this modeling gives an $A_\infty$-algebra which is unique up to $A_\infty$-isomorphism, for each homotopy type of cochain algebras.