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Licensed Unlicensed Requires Authentication Published by De Gruyter September 20, 2011

String topology of classifying spaces

David Chataur and Luc Menichi


Let G be a finite group or a compact connected Lie group and let BG be its classifying space. Let ℒBG ≔ map(S1, BG) be the free loop space of BG, i.e. the space of continuous maps from the circle S1 to BG. The purpose of this paper is to study the singular homology H*(ℒBG) of this loop space. We prove that when taken with coefficients in a field the homology of ℒBG is a homological conformal field theory. As a byproduct of our Main Theorem, we get a Batalin–Vilkovisky algebra structure on the cohomology H*(ℒBG). We also prove an algebraic version of this result by showing that the Hochschild cohomology HH*(S*(G), S*(G)) of the singular chains of G is a Batalin–Vilkovisky algebra.

Received: 2009-09-08
Revised: 2011-04-07
Published Online: 2011-09-20
Published in Print: 2012-08

©[2012] by Walter de Gruyter Berlin Boston