nLab nonabelian Poincaré duality

Idea

(…)

Related concepts

• nonabelian cohomology

• Ran space

• factorization algebra, topological chiral homology

• Weil conjecture on Tamagawa numbers

References

General

• Paolo Salvatore: Configuration spaces with summable labels, in Cohomological methods in homotopy theory (Bellaterra, 1998), Progr. Math. 196, Birkhäuser (2001) 375–395 [doi:10.1007/978-3-0348-8312-2_23, arXiv:math/9907073]

• Sadok Kallel: Particle Spaces on Manifolds and Generalized Poincaré Dualities, Quarterly J. Math. 52 1 (2001) 45–70 [doi:10.1093/qjmath/52.1.45, arXiv:math/9810067]

  (the abelian case reducing to ordinary Poincaré duality)

• Jacob Lurie, §3.8 in: Derived Algebraic Geometry VI: $\mathbb{E}_k$ Algebras [arXiv:0911.0018]

• Jacob Lurie, Def. 6 in: Nonabelian Poincaré Duality, Lecture 8 in: Tamagawa Numbers via Nonabelian Poincare Duality (282y) (2014) [pdf, pdf]

• Dennis Gaitsgory, Jacob Lurie: Nonabelian Poincaré Duality, section 3 in: Weil’s conjecture for function fields (2014-2017) [pdf, pdf]

• David Ayala, John Francis, §4 in: Factorization homology of topological manifolds, Journal of Topology 8 4 (2015) 1045-1084 [arXiv:1206.5522, doi:10.1112/jtopol/jtv028]

• Jacob Lurie, §5.5.6 in: Higher Algebra ($\sim$2017) [pdf]

• Jeremy Miller: Nonabelian Poincaré duality after stabilizing, Trans. Amer. Math. Soc. 367 (2015) 1969-1991 [doi:2015-367-03/S0002-9947-2014-06186-2, arXiv:1209.2773]

• Foling Zou, §4 in: A geometric approach to equivariant factorization homology and nonabelian Poincaré duality, Math. Z. 303 98 (2023) [arXiv:2008.08234, doi:10.1007/s00209-023-03253-2]

  (generalization to equivariant homotopy theory)

Further exposition:

• Dev Sinha, from 58:12 on in: The Fulton-MacPherson Operad and Recognition (2020) [video:YT]

In the generality of G-spaces/equivariant homotopy theory:

• Jeremy Hahn, Asaf Horev, Inbar Klang, Dylan Wilson, Foling Zou: Equivariant nonabelian Poincaré duality and equivariant factorization homology of Thom spectra [arXiv:2006.13348]

Application to Tanagawa numbers

The proof by Jacob Lurie and Dennis Gaitsgory via nonabelian Poincaré duality of the Weil conjecture on Tamagawa numbers was announced in

• Jacob Lurie, Tamagawa Numbers via Nonabelian Poincaré Duality, talk at FRG Chern-Simons workshop, Jan. 15-17, 2011

and details are at

• Jacob Lurie, Tamagawa Numbers via Nonabelian Poincare Duality (282y), lecture notes, 2014 (web)

Application to stable splitting of mapping spaces

Generalization via nonabelian Poincaré duality of stable splitting of mapping spaces from codomains which are $n$-fold suspensions to general n-connective spaces:

• Lauren Bandklayder, Stable splitting of mapping spaces via nonabelian Poincaré duality (arxiv:1705.03090)