String topology prospectra and Hochschild cohomology

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, S 0 [ 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.