Homotopy transfer and rational models for mapping spaces

Abstract

By using homotopy transfer techniques in the context of rational homotopy theory, we show that if $C$ is a coalgebra model of a space $X$, then the $A_{\infty }$-coalgebra structure in $H_{\ast }(X;\mathbb{Q})\cong H_{\ast }(C)$ induced by the higher Massey coproducts provides the construction of the Quillen minimal model of $X$. We also describe an explicit $L_{\infty }$-structure on the complex of linear maps $\mathrm{Hom}(H_{\ast }(X;\mathbb{Q}),{\pi }_{\ast }(\Omega Y)\otimes \mathbb{Q})$, where $X$ is a finite nilpotent CW-complex and $Y$ is a nilpotent CW-complex of finite type, modeling the rational homotopy type of the mapping space $\mathrm{map}(X,Y)$. As an application we give conditions on the source and target in order to detect rational $H$-space structures on the components.

21 pages. Final version. To appear in J. Homotopy Relat. Struct

References in corpus (1)

• Transferring homotopy commutative algebraic structures