nLab
equivariant stable cohomotopy

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Stable Homotopy theory

Representation theory

Contents

Idea

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.

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.

Properties

Equivariant stable π 3 𝕊\pi_3^{\mathbb{S}}

See at quaternionic Hopf fibration – Class in equivariant stable homotopy theory

Examples

Of the point: The Burnside ring

Proposition

(Burnside ring is equivariant stable cohomotopy of the point)

Let GG be a finite group, then its Burnside ring A(G)A(G) is isomorphic to the equivariant stable cohomotopy cohomology ring 𝕊 G(*)\mathbb{S}_G(\ast) of the point in degree 0.

A(G)𝕊 G(*). A(G) \overset{\simeq}{\longrightarrow} \mathbb{S}_G(\ast) \,.

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.

(equivariant) cohomologyrepresenting
spectrum
equivariant cohomology
of the point *\ast
cohomology
of classifying space BGB G
(equivariant)
ordinary cohomology
HZBorel equivariance
H G (*)H (BG,)H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})
(equivariant)
complex K-theory
KUrepresentation ring
KU G(*)R (G)KU_G(\ast) \simeq R_{\mathbb{C}}(G)
Atiyah-Segal completion theorem
R(G)KU G(*)compl.KU G(*)^KU(BG)R(G) \simeq KU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {KU_G(\ast)} \simeq KU(B G)
(equivariant)
complex cobordism cohomology
MUMU G(*)MU_G(\ast)completion theorem for complex cobordism cohomology
MU G(*)compl.MU G(*)^MU(BG)MU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {MU_G(\ast)} \simeq MU(B G)
(equivariant)
algebraic K-theory
K𝔽 pK \mathbb{F}_prepresentation ring
(K𝔽 p) G(*)R p(G)(K \mathbb{F}_p)_G(\ast) \simeq R_p(G)
Rector completion theorem
R 𝔽 p(G)K(𝔽 p) G(*)compl.(K𝔽 p) G(*)^Rector 73K𝔽 p(BG)R_{\mathbb{F}_p}(G) \simeq K (\mathbb{F}_p)_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {(K \mathbb{F}_p)_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Rector+completion+theorem">Rector 73</a>}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G)
(equivariant)
stable cohomotopy
K 𝔽 1Segal 74\mathbb{F}_1 \overset{\text{<a href="stable cohomotopy#StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement">Segal 74</a>}}{\simeq} SBurnside ring
𝕊 G(*)A(G)\mathbb{S}_G(\ast) \simeq A(G)
Segal-Carlsson completion theorem
A(G)Segal 71𝕊 G(*)compl.𝕊 G(*)^Carlsson 84𝕊(BG)A(G) \overset{\text{<a href="https://ncatlab.org/nlab/show/Burnside+ring+is+equivariant+stable+cohomotopy+of+the+point">Segal 71</a>}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Segal-Carlsson+completion+theorem">Carlsson 84</a>}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)

More explicitly, this means that the Burnside ring of a group GG is isomorphic to the colimit

A(G)lim V[S V,S V] G A(G) \simeq \underset{\longrightarrow}{\lim}_V [S^V,S^V]_G

over GG-representations in a complete G-universe, of GG-homotopy classes of GG-equivariant based continuous functions from the representation sphere S VS^V to itself (Greenlees-May 95, p. 8).

cohomologyequivariant cohomology
non-abelian cohomologycohomotopyequivariant cohomotopy
stable cohomologystable cohomotopyequivariant stable cohomotopy

References

Relation to Burnside ring

Relation to Segal-Carlsson completion theorem:

  • Czes Kosniowski, Equivariant cohomology and Stable Cohomotopy, Math. Ann. 210, 83-104 (1974) (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 (arXiv:math/0504051)

  • Noe Barcenas, Nonlinearity, Proper Actions and Equivariant Stable Cohomotopy (arXiv:1302.1712)

Discussion of unstable equivariant cohomotopy is in

  • James Cruickshank, Twisted homotopy theory and the geometric equivariant 1-stem, Topology and its Applications Volume 129, Issue 3, 1 April 2003, Pages 251-271 (doi:10.1016/S0166-8641(02)00183-9)

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)

Discussion of M-brane physics in terms of rational equivariant cohomotopy is in

Last revised on December 15, 2018 at 00:11:04. See the history of this page for a list of all contributions to it.