A geometric approach to equivariant factorization homology and nonabelian Poincaré duality

Abstract

Fix a finite group G and an n-dimensional orthogonal G-representation V. We define the equivariant factorization homology of a V-framed smooth G-manifold with coefficients in an $$\textrm{E}_V$$ -algebra using a two-sided bar construction, generalizing (Andrade, From manifolds to invariants of $$E_n$$ -algebras. PhD thesis, Massachusetts Institute of Technology, 2010; Kupers and Miller, Math Ann 370(1–2):209–269, 2018). This construction uses minimal categorical background and aims for maximal concreteness, allowing convenient proofs of key properties, including invariance of equivariant factorization homology under change of tangential structures. Using a geometrically-seen scanning map, we prove an equivariant version (eNPD) of the nonabelian Poincaré duality theorem due to several authors. The eNPD states that the scanning map gives a G-equivalence from the equivariant factorization homology to mapping spaces out the one-point compactification of the G-manifolds, when the coefficients are G-connected. For non-G-connected coefficients, when the G-manifolds have suitable copies of $$\mathbb {R}$$ in them, the scanning map gives group completions. This generalizes the recognition principle for V-fold loop spaces in Guillou and May (Algebr Geom Topol 17(6):3259–3339, 2017).