Monoid of self-equivalences and free loop spaces

Abstract:

Let $M$ be a simply-connected closed oriented $N$-dimensional manifold. We prove that for any field of coefficients $l\!k$ there exists a natural homomorphism of commutative graded algebras $\Gamma : H_* (\Omega \mbox {aut}_1 M) \to \mathbb H_{*}(M^{S^1})$ where $\mathbb H_*(M^{S^1})= H_{* +N}(M^{S^1})$ is the loop algebra defined by Chas and Sullivan. As usual $\mbox {aut}_1 X$ denotes the monoid of self-equivalences homotopic to the identity, and $\Omega X$ the space of based loops. When $l\!k$ is of characteristic zero, $\Gamma$ yields isomorphisms $H^{n+N}_{(1)}(M^{S^1}) \stackrel {\cong }{\to } (\pi _n(\Omega \mbox {aut}_1 M) \otimes l\!k)^\vee$ where $\bigoplus _{l=1}^\infty H^n_{(l)}(M^{S^1})$ denotes the Hodge decomposition on $H^* (M ^{S^1})$.