nLab nonabelian Poincaré duality

Idea

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Related concepts

• 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]

• Jacob Lurie, 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]

• 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]

The abelian case was discussed previously in

• 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]

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)