A Geometric Approach to Equivariant Factorization Homology and Nonabelian Poincaré Duality

Factorization homology is a homology theory on manifolds with coefficients in suitable $\mathrm{E}_n$-algebras. In this paper, we use the minimal categorical background and maximal concreteness to study equivariant factorization homology in the $V$-framed case. We work with a finite group $G$ and an $n$-dimensional orthogonal $G$-representation $V$. The main results are: \begin{enumerate} \item We construct a $G\mathrm{Top}$-enriched category $\mathrm{Mfld}^{\mathrm{fr}_{V}}_{n}$. Its objects are $V$-framed $G$-manifolds of dimension $n$. The endomorphism operad of the object $V$ is equivalent to the little $V$-disk operad. \item With this category, we define the equivariant factorization homology $\displaystyle\int_MA$ by a monadic bar construction. \item We prove the nonabelian Poincar\'e duality theorem using a geometrically-seen scanning map, which establishes a weak $G$-equivalence between $\displaystyle\int_MA$ and $\mathrm{Map}_*(M^+, \mathbf{B}^VA)$. \end{enumerate} Here, $M$ is a $V$-framed manifold, and $M^+$ is its one-point compactification. In the language of Guillou-May \cite{GM17}, the coefficient $A$ is an algebra over the little $V$-disks operad and $\mathbf{B}^VA$ is the $V$-fold deloop of $A$.