Let be a path-connected closed oriented -dimensional smooth manifold and let be a principal ideal domain. By Chas and Sullivan, the shifted free loop space homology of , is a Batalin–Vilkovisky algebra. Let be a topological group such that is a classifying space of . Denote by the (normalized) singular chains on . Suppose that is discrete or path-connected. We show that there is a Van Den Bergh type isomorphism
Therefore, the Gerstenhaber algebra is a Batalin–Vilkovisky algebra and we have a linear isomorphism
This linear isomorphism is expected to be an isomorphism of Batalin–Vilkovisky algebras. We also give a new characterization of Batalin–Vilkovisky algebra in terms of the derived bracket.
Cite this article
Luc Menichi, Van den Bergh isomorphisms in string topology. J. Noncommut. Geom. 5 (2011), no. 1, pp. 69–105DOI 10.4171/JNCG/70