  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.
