Equivariant nonabelian Poincar\'e duality and equivariant factorization homology of Thom spectra

TL;DR

This paper develops a framework for equivariant factorization homology of Thom spectra, proves an equivariant nonabelian Poincaré duality theorem, and computes examples like Real THR of certain spectra, advancing understanding of equivariant homology theories.

Contribution

It introduces a new approach to equivariant factorization homology of Thom spectra and establishes a nonabelian Poincaré duality theorem in the equivariant setting.

Findings

Computed Real THR of $MU_\mathbb{R}$, $H\underline{\mathbb{F}}_2$, and $H\underline{\mathbb{Z}}_{(2)}$.

Described genuine equivariant factorization homology of Thom spectra.

Proved an equivariant nonabelian Poincaré duality theorem relating factorization homology and mapping spaces.

Abstract

In this paper, we study genuine equivariant factorization homology and its interaction with equivariant Thom spectra, which we construct using the language of parametrized higher category theory. We describe the genuine equivariant factorization homology of Thom spectra, and use this description to compute several examples of interest. A key ingredient for our computations is an equivariant nonabelian Poincar\'e duality theorem, in which we prove that factorization homology with coefficients in a $G$-space is given by a mapping space. We compute the Real topological Hochschild homology ($THR$) of the Real bordism spectrum $MU_\mathbb{R}$ and of the equivariant Eilenberg--MacLane spectra $H\underline{\mathbb{F}}_2$ and $H\underline{\mathbb{Z}}_{(2)}$, as well as factorization homology of the sphere $S^{2\sigma}$ with coefficients in these Eilenberg--MacLane spectra. In Appendix B, Jeremy Hahn and Dylan Wilson compute $THR(H\underline{\mathbb{Z}})$.