String topology of classifying spaces

David Chataur; Luc Menichi

String topology of classifying spaces

Algebraic Topology 2009-06-01 v3 Quantum Algebra

Authors: David Chataur , Luc Menichi

Abstract

Let $G$ be a finite group or a compact connected Lie group and let $BG$ be its classifying space. Let $\mathcal{L}BG:=map(S^1,BG)$ be the free loop space of $BG$ i.e. the space of continuous maps from the circle $S^1$ to $BG$ . The purpose of this paper is to study the singular homology $H_*(\mathcal LBG)$ of this loop space. We prove that when taken with coefficients in a field the homology of $\mathcal LBG$ is a homological conformal field theory. As a byproduct of our main theorem, we get a Batalin-Vilkovisky algebra structure on the cohomology $H^*(\mathcal LBG)$ . 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.

Keywords

hochschild cohomology cohomology algebraic topology

Cite

@article{arxiv.0801.0174,
  title  = {String topology of classifying spaces},
  author = {David Chataur and Luc Menichi},
  journal= {arXiv preprint arXiv:0801.0174},
  year   = {2009}
}

Comments

53 pages. Section 3 on Props and fields theories rewritten. Section 4 expanded in new sections 4, 5, 6 and 7, to fix orientation problems, finite groups case detailed in section 7. Appendix on signs added. The rest of the sections almost unchanged. Some slight improvements on some results. For example, the BV-algebra is valid over any principal ideal domain

String topology of classifying spaces

Abstract

Keywords

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