A cocyclic construction of \(S^1\)-equivariant homology and application to string topology (Q6594007)

Chas and Sullivan constructed a gravity algebra structure on the \(S^1\)-equivariant homology of the free loop space \(LM\) for a closed manfiold \(M\). This gravity algebra on \(\mathrm{H}^{S^1}_*(LM)\) is induced by the Batalin-Vilkovisky algebra structure on the homology of \(LM\). \textit{K. Irie} has shown in [Int. Math. Res. Not. 2018, No. 15, 4602--4674 (2018; Zbl 1414.55005)] that the BV algebra structure on loop space homology has a chain level refinement. In the present article the author shows a similar result for the \(S^1\)-equivariant homology of \(LM\) with its gravity algebra structure.\N\NMore precisely, it is shown in Theorem 7.6 that for a closed oriented smooth manifold \(M\) there is a chain complex \(\widetilde{O}_M^{\mathrm{cyc}}\) whose homology is the \(S^1\)-equivariant homology of \(LM\) and such that this chain complex admits an action of the chain model of the gravity operad constructed by \textit{B. C. Ward} [J. Noncommut. Geom. 10, No. 4, 1403--1464 (2016; Zbl 1375.18056)]. This structure lifts the gravity algebra structure on \(\mathrm{H}^{S^1}_*(LM)\). Moreover, there is a morphism of \(\widetilde{O}_M^{\mathrm{cyc}}\) to a complex that computes the cyclic homology of the de Rham cochains on \(M\) and this morphism accepts the algebra structure over the chain model of the gravity operad.\N\NIn the beginning of the paper (Theorem 3.1) the author shows that for an \(S^1\)-space \(X\) the cosimplical complex \(\{S_*(X\times \Delta_k)\}_{k\geq 0}\) can be made into a \textit{cocyclic chain complex} such that the cyclic homology of \(\{S_*(X\times \Delta^k)\}_{k\geq 0}\) and the \(S^1\)-equivariant homology of \(X\) are isomorphic in a natural way. This is related to Jones' well-known result in [\textit{J. D. S. Jones}, Invent. Math. 87, 403--423 (1987; Zbl 0644.55005)].