String topology prospectra and Hochschild cohomology

2007/10/07 by Kate Gruher, Craig Westerland, Gruher, Kate +1
Mathematics · Physics and Astronomy · #16E40 #55P25 #55P35 #55R10 #55R12 #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #math.AT #math.QA #msc:16E40 #msc:55P25 #msc:55P35 #msc:55R10 #msc:55R12

paper · pdf · doi:10.48550/arxiv.0710.1445

19 pages. Comments welcome. References added, some statements clarified

openalex publication_date 2007/10/07 · arxiv created 2007/11/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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, 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.

Related

• String topology of classifying spaces
• A homotopy theoretic realization of string topology
• The string topology BV algebra, Hochschild cohomology and the Goldman bracket on surfaces
• Equivariant operads, string topology, and Tate cohomology
• The bv algebra in string topology of classifying spaces