Rational homotopy theory of function spaces and Hochschild cohomology

Abstract

Given a map $f:X\to Y$ of simply connected spaces of finite type such. The space of based loops at $f$ of the space of maps between $X$ and $Y$ is denoted by ${\Omega }_{f}Map(X,Y)$. For $n>0$, we give a model categorical interpretation of the existence (in functorial way) of an injective map of $\mathbb{Q}$-vector spaces ${\pi }_n{\Omega }_{f}Map(X,Y_{\mathbb{Q}})\to H{H}^{-n}(C^{\ast }(Y),C^{\ast }(X{)}_f)$, where $H{H}^{\ast }$ is the (negative) Hochschild cohomology and $C^{\ast }(X{)}_f$ is the rational cochain complex associated to $X$ equipped with a structure of $C^{\ast }(Y)$-differential graded bimodule via the induced map of differential graded algebras $f^{\ast }:C^{\ast }(Y)\to C^{\ast }(X)$. Moreover, we identifiy the image in presice way by using the Hodge filtration on Hochschild cohomology. In particular, when $X=Y$, we describe the fundamental group of the identity component of the monoid of self equivalence of a (rationalization of) space $X$ i.e., ${\pi }_{1}Aut(X_{\mathbb{Q}}{)}_{id}$.