group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
geometric representation theory
representation, 2-representation, ∞-representation
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Be?linson-Bernstein localization?
The equivariant generalized cohomology theory which is represented by the equivariant sphere spectrum may also be called equivariant stable cohomotopy, as it is the equivariant stable homotopy theory version of stable cohomotopy, hence of cohomotopy. This is to be thought of as the first order Goodwillie approximation of plain (“unstable”) equivariant cohomotopy.
Just as the plain sphere spectrum is a distinguished object of plain stable homotopy theory, so the equivariant sphere spectrum is distinguished in equivariant stable homotopy theory and hence so is equivariant stable cohomotopy theory.
The following is known as the Barratt-Priddy-Quillen theorem:
(stable cohomotopy is K-theory of FinSet)
Let FinSet be a skeleton of the category of finite sets, regarded as a permutative category. Then the K-theory of this permutative category
is represented by the sphere spectrum, hence is stable cohomotopy.
This is due to Barratt-Priddy 72 reproved in Segal 74, Prop. 3.5. See also Priddy 73, Glasman 13.
(stable cohomotopy as algebraic K-theory over the field with one element)
Notice that for a field, the K-theory of a permutative category of its category of modules is its algebraic K-theory (see this example)
Now (pointed) finite sets may be regarded as the modules over the “field with one element” (see there):
If this is understood, example says that stable cohomotopy is the algebraic K-theory of the field with one element:
This perspective is highlighted for instance in (Deitmar 06, p. 2, Guillot 06, Mahanta 17, Dundas-Goodwillie-McCarthy 13, II 1.2, Morava, Connes-Consani 16).
The perspective that the K-theory over should be stable Cohomotopy has been highlighted in (Deitmar 06, p. 2, Guillot 06, Mahanta 17, Dundas-Goodwillie-McCarthy 13, II 1.2, Morava, Connes-Consani 16). Generalized to equivariant stable homotopy theory, the statement that equivariant K-theory over should be equivariant stable Cohomotopy is discussed in Chu-Lorscheid-Santhanam 10, 5.3.
See at quaternionic Hopf fibration – Class in equivariant stable homotopy theory
(Burnside ring is equivariant stable cohomotopy of the point)
Let be a finite group, then its Burnside ring is isomorphic to the equivariant stable cohomotopy cohomology ring of the point in degree 0.
This is due to Segal 71, a detailed proof is given by tom Dieck 79, theorem 8.5.1. See also Lück 05, theorem 1.13, tom Dieck-Petrie 78.
More explicitly, this means that the Burnside ring of a group is isomorphic to the colimit
over -representations in a complete G-universe, of -homotopy classes of -equivariant based continuous functions from the representation sphere to itself (Greenlees-May 95, p. 8).
flavours of Cohomotopy cohomology theory | cohomology (full or rational) | equivariant cohomology (full or rational) |
---|---|---|
non-abelian cohomology | Cohomotopy (full or rational) | equivariant Cohomotopy |
twisted cohomology (full or rational) | twisted Cohomotopy | twisted equivariant Cohomotopy |
stable cohomology (full or rational) | stable Cohomotopy | equivariant stable Cohomotopy |
differential cohomology | differential Cohomotopy | equivariant differential cohomotopy |
persistent cohomology | persistent Cohomotopy | persistent equivariant Cohomotopy |
Relation to Burnside ring:
Graeme Segal, Equivariant stable homotopy theory, In Actes du Congrès International des Math ématiciens (Nice, 1970), Tome 2 , pages 59–63. Gauthier-Villars, Paris, 1971 (pdf)
Tammo tom Dieck, T. Petrie, Geometric modules over the Burnside ring, Invent. Math. 47 (1978) 273-287 (pdf)
Tammo tom Dieck, Transformation Groups and Representation Theory, Springer 1979
Tammo tom Dieck, Section II.8 of: Transformation Groups, de Gruyter 1987 (doi:10.1515/9783110858372)
Relation to Segal-Carlsson completion theorem:
Czes Kosniowski, Equivariant cohomology and Stable Cohomotopy, Math. Ann. 210, 83-104 (1974) (doi:10.1007/BF01360033 pdf)
Erkki Laitinen, On the Burnside ring and stable cohomotopy of a finite group, Mathematica Scandinavica Vol. 44, No. 1 (August 30, 1979), pp. 37-72 (jstor:24491306, pdf)
Gunnar Carlsson, Equivariant Stable Homotopy and Segal’s Burnside Ring Conjecture, Annals of Mathematics Second Series, Vol. 120, No. 2 (Sep., 1984), pp. 189-224 (jstor:2006940, pdf)
Wolfgang Lück, The Burnside Ring and Equivariant Stable Cohomotopy for Infinite Groups, Pure Appl. Math. Q. 1 (2005), no. 3, Special Issue: In memory of Armand Borel. Part 2, 479–541 (arXiv:math/0504051)
Noe Barcenas, Nonlinearity, Proper Actions and Equivariant Stable Cohomotopy (arXiv:1302.1712)
A lift of Seiberg-Witten invariants to classes in circle group-equivariant stable cohomotopy is discussed in
Stefan Bauer, Mikio Furuta A stable cohomotopy refinement of Seiberg-Witten invariants: I (arXiv:math/0204340)
Stefan Bauer, A stable cohomotopy refinement of Seiberg-Witten invariants: II (arXiv:math/0204267)
Christian Okonek, Andrei Teleman, Cohomotopy Invariants and the Universal Cohomotopy Invariant Jump Formula, J. Math. Sci. Univ. Tokyo 15 (2008), 325-409 (pdf)
The identification of stable cohomotopy with the K-theory of the permutative category of finite set is due to
Michael Barratt, Stewart Priddy, On the homology of non-connected monoids and their associated groups, Commentarii Mathematici Helvetici, December 1972, Volume 47, Issue 1, pp 1–14 (doi:10.1007/BF02566785)
Graeme Segal, Categories and cohomology theories, Topology vol 13, pp. 293-312, 1974 (pdf)
see also
Stewart Priddy, Transfer, symmetric groups, and stable homotopy theory, in Higher K-Theories, Springer, Berlin, Heidelberg, 1973. 244-255 (pdf)
Saul Glasman, The multiplicative Barratt-Priddy-Quillen theorem and beyond, talk 2013 (pdf)
The resulting interpretation of stable cohomotopy as algebraic K-theory over the field with one element is amplified in the following texts:
Bjørn Dundas, Thomas Goodwillie, Randy McCarthy, chapter II, section 1.2 of The local structure of algebraic K-theory, Springer 2013 (pdf)
Anton Deitmar, Remarks on zeta functions and K-theory over (arXiv:math/0605429)
Pierre Guillot, Adams operations in cohomotopy (arXiv:0612327)
Snigdhayan Mahanta, G-theory of -algebras I: the equivariant Nishida problem, J. Homotopy Relat. Struct. 12 (4), 901-930, 2017 (arXiv:1110.6001)
Chenghao Chu, Oliver Lorscheid, Rekha Santhanam, Sheaves and K-theory for -schemes, Advances in Mathematics, Volume 229, Issue 4, 1 March 2012, Pages 2239-2286 (arxiv:1010.2896)
see also
Jack Morava, Some background on Manin’s theorem (pdf, MoravaSomeBackground.pdf)
Alain Connes, Caterina Consani, Absolute algebra and Segal’s Gamma sets, Journal of Number Theory 162 (2016): 518-551 (arXiv:1502.05585)
John D. Berman, p. 92 of: Categorified algebra and equivariant homotopy theory, PhD thesis 2018 (arXiv:1805.08745)
Proof that equivariant framed bordism homology theory is co-represented by the equivariant sphere spectrum:
Discussion for M-brane physics:
Last revised on June 12, 2021 at 09:20:42. See the history of this page for a list of all contributions to it.