Hochschild cohomology of Sullivan algebras and mapping spaces between manifolds

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.