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. 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.

Properties

As equivariant algebraic K-theory over 𝔽 1\mathbb{F}_1

The following is known as the Barratt-Priddy-Quillen theorem:

Proposition

(stable cohomotopy is K-theory of FinSet)

Let 𝒞=\mathcal{C} = FinSet be a skeleton of the category of finite sets, regarded as a permutative category. Then the K-theory of this permutative category

K(FinSet)𝕊 K(FinSet) \;\simeq\; \mathbb{S}

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.

Remark

(stable cohomotopy as algebraic K-theory over the field with one element)

Notice that for FF a field, the K-theory of a permutative category of its category of modules FModF Mod is its algebraic K-theory KFK F (see this example)

KFK(FMod). K F \;\simeq\; K(F Mod) \,.

Now (pointed) finite sets may be regarded as the modules over the “field with one element𝔽 1\mathbb{F}_1 (see there):

𝔽 1Mod=FinSet */ \mathbb{F}_1 Mod \;=\; FinSet^{\ast/}

If this is understood, example says that stable cohomotopy is the algebraic K-theory of the field with one element:

𝕊K𝔽 1. \mathbb{S} \;\simeq\; K \mathbb{F}_1 \,.

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 K𝔽 1K \mathbb{F}_1 over 𝔽 1\mathbb{F}_1 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 K G𝔽 1 K_G \mathbb{F}_1 over 𝔽 1\mathbb{F}_1 should be equivariant stable Cohomotopy is discussed in Chu-Lorscheid-Santhanam 10, 5.3.

(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
K𝔽 1Segal 74K \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)

Examples

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

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

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
K𝔽 1Segal 74K \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).

flavours of
Cohomotopy
cohomology theory
cohomology
(full or rational)
equivariant cohomology
(full or rational)
non-abelian cohomologyCohomotopy
(full or rational)
equivariant Cohomotopy
twisted cohomology
(full or rational)
twisted Cohomotopytwisted equivariant Cohomotopy
stable cohomology
(full or rational)
stable Cohomotopyequivariant stable Cohomotopy

References

Relation to Burnside ring

Relation to Burnside ring:

Relation to Segal-Carlsson completion theorem

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)

As equivariant K-theory over the field with one element

The identification of stable cohomotopy with the K-theory of the permutative category of finite set is due to

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:

see also

In M-brane charge quantization

Discussion for M-brane physics:

Last revised on October 4, 2019 at 09:07:54. See the history of this page for a list of all contributions to it.