JournalsjncgVol. 5 , No. 1DOI 10.4171/jncg/70

Van den Bergh isomorphisms in string topology

  • Luc Menichi

    Université d’Angers, France
Van den Bergh isomorphisms in string topology cover

Abstract

Let MM be a path-connected closed oriented dd-dimensional smooth manifold and let k\mathbb{k} be a principal ideal domain. By Chas and Sullivan, the shifted free loop space homology of MM, H+d(LM)H_{*+d}(\mathrm{L}\mkern-1.5mu M ) is a Batalin–Vilkovisky algebra. Let GG be a topological group such that MM is a classifying space of GG. Denote by S(G)S_*(G) the (normalized) singular chains on GG. Suppose that GG is discrete or path-connected. We show that there is a Van Den Bergh type isomorphism

HHp(S(G),S(G))HHp+d(S(G),S(G)).\mathrm{HH}^{-p}(S_*(G),S_*(G))\cong \mathrm{HH}_{p+d}(S_*(G),S_*(G)).

Therefore, the Gerstenhaber algebra HH(S(G),S(G))\mathrm{HH}^{*}(S_*(G),S_*(G)) is a Batalin–Vilkovisky algebra and we have a linear isomorphism

HH(S(G),S(G))H+d(LM).\mathrm{HH}^{*}(S_*(G),S_*(G))\cong H_{*+d}(\mathrm{L}\mkern-1.5mu M ).

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.