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Journal of Topology Advance Access originally published online on November 26, 2008
Journal of Topology 2008 1(4):837-856; doi:10.1112/jtopol/jtn025
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© 2008 London Mathematical Society

String topology prospectra and Hochschild cohomology

Kate Gruher

Stanford University, Department of Mathematics, Building 380, Stanford, CA 94305, USA kagruher@stanford.edu

Craig Westerland

Department of Mathematics and Statistics, University of Melbourne, Parkville, Victoria, 3010, Australia C.Westerland@ms.unimelb.edu.au


  Abstract

We study string topology for classifying spaces of connected compact Lie groups, drawing connections with Hochschild cohomology and equivariant homotopy theory. First, for a compact Lie group G, we show that the string topology prospectrum LBG– TBG is equivalent to the homotopy fixed-point prospectrum for the conjugation action of G on itself, S0[G]hG. Dually, we identify LBG-ad with the homotopy orbit spectrum (DG)hG, and study ring and co-ring structures on these spectra. Finally, we show that in homology, these products may be identified with the Gerstenhaber cup product in the Hochschild cohomology of C*(BG) and C*(G), respectively. These, in turn, are isomorphic via Koszul duality.

Received November 9, 2007.

2000 Mathematics Subject Classification 55R10, 55R12, 55P35, 55P25, 16E40

This material is based upon work partially supported by the National Science Foundation under agreement no. DMS-0705428.


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