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Equivariant nonabelian Poincaré duality and equivariant factorization homology of Thom spectra

Jeremy Hahn, Asaf Horev, Inbar Klang, Dylan Wilson, Foling Zou

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 Poincaré 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 $S^{2σ}$ with coefficients in these Eilenberg--MacLane spectra. In Appendix B, Jeremy Hahn and Dylan Wilson compute $THR(H\underline{\mathbb{Z}})$.

Equivariant nonabelian Poincaré duality and equivariant factorization homology of Thom spectra

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 Poincaré duality theorem, in which we prove that factorization homology with coefficients in a -space is given by a mapping space. We compute the Real topological Hochschild homology () of the Real bordism spectrum and of the equivariant Eilenberg--MacLane spectra and , as well as factorization homology of the sphere with coefficients in these Eilenberg--MacLane spectra. In Appendix B, Jeremy Hahn and Dylan Wilson compute .

Paper Structure

This paper contains 51 sections, 64 theorems, 216 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Theorem 1.
  3. Corollary 2.
  4. Theorem 3.
  5. Theorem 4.
  6. Theorem 5.
  7. Theorem 6.
  8. Structure of the paper.
  9. Foundations.
  10. Notation.
  11. Acknowledgments.
  12. From topological functors to G-symmetric monoidal functors
  13. Preliminaries on $G$-$\infty$-categories
  14. The $G$-$\infty$-category of topological $G$-objects
  15. Constructing a topological category over $\ul{\mathbf{Fin}}^G_*$.
  16. ...and 36 more sections

Key Result

Theorem 2.2.1

Let $N(\ul\mathscr{C}^\otimes)$ denote the coherent nerve of $\ul\mathscr{C}^\otimes$ (as in HTT). Then is a $G$-symmetric monoidal $G$-$\infty$-category.

Figures (3)

  • Figure 1: Attaching a $G$-handle $D(E) \times_U D(E')$ of a critical point $x \in U$ with stabilizer $H$ and orbit $U=\left\{{x,gx}\right\}$
  • Figure 2: Notation
  • Figure 3: Illustration

Theorems & Definitions (152)

  • Definition 2.1.1
  • Example 2.1.3
  • Remark 2.1.4
  • Remark 2.1.5
  • Example 2.1.7
  • Definition 2.1.8: Parametrized_algebra, combination of def. 2.2.3. and prop. 2.2.6
  • Example 2.1.10
  • Remark 2.1.14
  • Theorem 2.2.1
  • Proposition 2.2.2
  • ...and 142 more