The factorization theory of Thom spectra and twisted nonabelian Poincaré duality

Abstract

We give a description of the factorization homology and $E_n$ topological Hochschild cohomology of Thom spectra arising from $n$-fold loop maps $f: A \to BO$, where $A = \Omega^n X$ is an $n$-fold loop space. We describe the factorization homology $\int_M Th(f)$ as the Thom spectrum associated to a certain map $\int _M A \to BO$. When $M$ is framed and $X$ is $(n-1)$-connected, this spectrum is equivalent to a Thom spectrum of a virtual bundle over the mapping space $Map_c(M,X)$; in general, this is a Thom spectrum of a virtual bundle over a certain section space. This can be viewed as a twisted form of the non-abelian Poincar\'e duality theorem of Segal, Salvatore, and Lurie, which occurs when $f: A \to BO$ is nullhomotopic. This result also generalizes the results of Blumberg-Cohen-Schlichtkrull on the topological Hochschild homology of Thom spectra, and of Schlichtkrull on higher topological Hochschild homology of Thom spectra. We use this to calculate the factorization homology of some Thom spectra, including the classical cobordism spectra, spectra arising from systems of groups, and the Eilenberg-MacLane spectra $H \mathbb{Z}/p$, $H\mathbb{Z}_{(p)}$, and $H\mathbb{Z}$. We build upon the description of the factorization homology of Thom spectra to study the ($n=1$ and higher) topological Hochschild cohomology of Thom spectra, which enables calculations and a description in terms of sections of a parametrized spectrum. If $X$ is a closed manifold, Atiyah duality for parametrized spectra allows us to deduce a duality between $E_n$ topological Hochschild homology and $E_n$ topological Hochschild cohomology, recovering string topology operations when $f$ is nullhomotopic. In conjunction with the higher Deligne conjecture, this gives $E_{n+1}$ structures on a certain family of Thom spectra, which were not previously known to be ring spectra.