Hochschild cohomology of Sullivan algebras and mapping spaces between manifolds

Document Type : Research Article


Department of Mathematics and Statistical Sciences‎, ‎Botswana International University of Science and Technology.


‎Let $e‎: ‎N^n \rightarrow M ^m$ be an embedding of closed‎, ‎oriented manifolds of dimension $n$ and $m$ respectively‎. ‎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 ‎$ H_*(e)_{!}‎: ‎H_*( LM‎, ‎\mathbb{Q}) \rightarrow H_*( L_NM‎, ‎\mathbb{Q})$ using Hochschild cohomology and study its properties‎. ‎In particular we extend a result of F'elix on the injectivity of the map induced by $ \aut_1M \rightarrow \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.


Main Subjects

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