Spaces of self-equivalences and free loops spaces

Abstract

Let M be a simply-connected closed oriented N-dimensional manifold. We prove that for any field of coefficients there exists a natural homomorphism of commutative graded algebras $Ψ: H_\ast (Ω{aut}_1 M) \to H_{\ast +N}(M^{S^1})$ where $H_\ast (M^{S^1})$ is the loop algebra defined by Chas-Sullivan. As usual ${aut}_1 X$ (resp. $ΩX$) denotes the monoid of the self-equivalences homotopic to the identity map (resp. the space of based loops) of the space X. Moreover, if $\bk$ is of characteristic zero, $Ψ$ yields isomorphisms $π_n(Ω{aut}_1 M) \otimes \bk \cong \hH^{n+N}_{(1)}$ where $\displaystyle \oplus_{l=1}^\infty \hH^n_{(l)}$ denotes the Hodge decomposition on $H^\ast (M ^{S^1})$.