A reduction of the string bracket to the loop product (Q6614582)

This article deals with the Chas-Sullivan string bracket on the \({S}^1\)-equivariant homology of the free loop space of a space \(M\). For a closed oriented manifold \(M\) the \textit{string bracket} is a map \N\[\N[\cdot,\cdot]\colon \mathrm{H}_{\bullet}^{{S}^1}(LM) \otimes \mathrm{H}_{\bullet}^{{S}^1}(LM) \to \mathrm{H}_{\bullet+2-n}^{{S}^1}(LM) \N\]\Nwhere \(n = \mathrm{dim}(M)\), which endows \(\mathrm{H}_{\bullet}^{{S}^1}(LM)\) with a graded Lie algebra structure. As one of the main results of the article (Theorem 2.15) the authors prove the following.\N\NIf \(M\) is a simply connected closed BV-exact manifold and \({K}\) is a field of characteristic zero, then the string bracket is equal to the composition \N\[\N\mathrm{H}_{\bullet}^{{S}^1}(LM; {K})^{\otimes 2} \xrightarrow{} \mathrm{H}_{\bullet}(LM;K)^{\otimes 2} \xrightarrow{\wedge_{\mathrm{CS}}} \mathrm{H}_{\bullet}(LM;K) \xrightarrow{\Delta'} \mathrm{H}_{\bullet}^{{S}^1}(LM; {K}) \N\]\Nwhere \(\wedge_{\mathrm{CS}}\) is the Chas-Sullivan product and \(\Delta'\colon \mathrm{H}_{\bullet}(LM;{K})\to \mathrm{H}_{\bullet}^{{S}^1}(LM;{K})\) is the composition of the BV-operator \(\Delta\colon \mathrm{H}_{\bullet}(LM;{K})\to \mathrm{H}_{\bullet+1}(LM;{K})\) with a map to \(\mathrm{H}_{\bullet}^{{S}^1}(LM;{K})\). The authors call this a a reduction of the string bracket to the loop product.\N\NThe notion of \textit{BV-exactness} is defined in a quite general setting. For a simply connected manifold BV-exactness means that the map given by the BV operator on reduced homology \(\widetilde{\Delta}\colon \widetilde{\mathrm{H}}_{\bullet}(LM) \to \widetilde{\mathrm{H}}_{\bullet+1}(LM)\) satisfies \(\mathrm{ker}(\widetilde{\Delta}) = \mathrm{im}(\widetilde{\Delta})\). The authors further study in what situations BV-exactness holds. Among other results they prove in Corollary 2.13 that simply connected manifolds whose rational cohomology ring is generated by one element as well as classifying spaces of compact connected Lie groups are BV-exact.\N\NThe result mentioned above (Theorem 2.15) is actually a special case of a more general theorem (Theorem 2.7) where the authors study the algebraic descriptions of the ordinary and the \({S}^1\)-equivariant homology of the free loop space by the Hochschild homology as well as by the cyclic homology of the polynomial de Rham algebra of a space. The authors also prove in this more general situation that the string bracket can be reduced to the loop product. In particular the applications to string topology do not only work for simply connected manifolds but for all Gorenstein spaces including classifying spaces of compact connected Lie groups.\N\NIn the end of the paper the authors also give example computations of the string bracket for a nonformal space which is BV-exact and they provide an example of a space which is not BV-exact.