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 $E_V$-algebra using a two-sided bar construction, generalizing [And10, KM18]. 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 Poincare 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 R in them, the scanning map gives group completions. This generalizes the recognition principle for V -fold loops spaces in [GM17].

arXiv comment46 pages; major changes from version 1, including rewriting the introduction and adding new sections 3.4 and 4.7

References in corpus (8)

• Integrating quantum groups over surfaces
• Higher enveloping algebras
• The geometry of the cyclotomic trace
• Factorization homology and calculus à la Kontsevich Soibelman
• Genuine equivariant factorization homology
• Equivariant nonabelian Poincaré duality and equivariant factorization homology of Thom spectra
• Unital operads, monoids, monads, and bar constructions
• Notes on equivariant bundles

Cited by in corpus (2)

• Equivariant nonabelian Poincaré duality and equivariant factorization homology of Thom spectra
• Unital operads, monoids, monads, and bar constructions