A tower connecting gauge groups to string topology

Abstract

Abstract. We develop a variant of calculus of functors, and use it to relate the gauge group G(P) of a principal bundle P over M to the Thom ring spectrum (P Ad ) −T M . If P has contractible total space, the resulting Thom ring spectrum is LM −T M , which plays a central role in string topology. Cohen and Jones have recently observed that, in a certain sense, (P Ad ) −T M is the linear approximation of G(P). We prove an extension of that relationship by demonstrating the existence of higher-order approximations and calculating them explicitly. This also generalizes calculations done by Arone in [Aro99].

Coassembly is a homotopy limit map

“…We could alternatively describe F % (X → B) as the spectrum of sections of a parametrized spectrum over X whose fiber over x is F ((x∐B) → B). See [WW95b], [Wil00], [CK09], [Mal15], and [Mal17] for more details and other explicit constructions of the coassembly map.…”