Rational homotopy theory of function spaces and Hochschild cohomology

TL;DR

This paper explores the connection between the rational homotopy groups of function spaces and Hochschild cohomology, providing a categorical model and explicit identification of the image, especially for self-equivalences.

Contribution

It offers a model categorical interpretation of the rational homotopy groups of function spaces in terms of Hochschild cohomology and identifies the image using the Hodge filtration.

Findings

Established an injective map between homotopy groups and Hochschild cohomology.

Identified the image of this map via the Hodge filtration.

Described the fundamental group of the space of self-equivalences for rationalized spaces.

Abstract

Given a map $f: X\rightarrow Y$ 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 $\pi_{n} \Omega_{f}Map(X,Y_{\mathbb{Q}}) \rightarrow HH^{-n}(C^{\ast}(Y),C^{\ast}(X)_{f})$, where $HH^{\ast}$ 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 $f^{\ast}: C^{\ast}(Y)\rightarrow C^{\ast}(X)$. 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., $\pi_{1} Aut(X_{\mathbb{Q}})_{id}$.