Hochschild cohomology of Sullivan algebras and mapping spaces between manifolds

TL;DR

This paper explores the relationship between the homology of free loop spaces on manifolds and the Hochschild cohomology of Sullivan algebras, introducing a shriek map and extending results on rational homotopy groups.

Contribution

It defines a new shriek map between loop space homologies using Hochschild cohomology and extends Félix's result on the injectivity of certain induced maps on rational homotopy groups.

Findings

Defined a shriek map e_! between homologies of loop spaces.

Extended Félix's injectivity result to cases where manifolds have the same dimension.

Analyzed properties of the shriek map in the context of Sullivan algebras.

Abstract

Let $e: N^n \rightarrow M^m $ be an embedding into a compact manifold $M$. We study the relationship between the homology of the free loop space $LM$ on $M$ and of the space $L_NM$ of loops of $M$ based in $N$ and define a shriek map $ e_{!}: H_*( LM, \mathbb{Q}) \rightarrow H_*( L_NM, \mathbb{Q})$ using Hochschild cohomology and study its properties. We also extend a result of F\'elix on the injectivity of the induced map $ \mathrm{aut}_1M \rightarrow \mathrm{map}(N, M; f ) $ on rational homotopy groups when $M$ and $N$ have the same dimension and $ f: N\rightarrow M $ is a map of non zero degree.