nLab equivariant stable cohomotopy

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 $\mathbb{F}_1$

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

Proposition

(stable cohomotopy is K-theory of FinSet)

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

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

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 $F$ a field, the K-theory of a permutative category of its category of modules $F Mod$ is its algebraic K-theory $K F$ (see this example)

$$K F \;\simeq\; K(F Mod) \,.$$

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

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

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

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

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

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

Examples

Equivariant stable $\pi_3^{\mathbb{S}}$

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

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

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

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

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

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

Related concepts

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

• Burnside ring

• Segal conjecture, Sullivan conjecture

References

Relation to Burnside ring

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

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

• 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 $\mathbb{F}_1$ (arXiv:math/0605429)

• Pierre Guillot, Adams operations in cohomotopy (arXiv:0612327)

• Snigdhayan Mahanta, G-theory of $\mathbb{F}_1$-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 $\mathbb{F}_1$-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 $K(\mathbb{F}_1) \sim \mathbb{S}$ (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)

Relation to equivariant cobordism theory

Proof that equivariant framed bordism homology theory is co-represented by the equivariant sphere spectrum:

• Czes Kosniowski, Equivariant Stable Homotopy and Framed Bordism, Transactions of the American Mathematical Society Vol. 219 (1976), pp. 225-234 (jstor:1997591)

In M-brane charge quantization

Discussion for M-brane physics:

• John Huerta, Hisham Sati, Urs Schreiber, Real ADE-equivariant (co)homotopy and Super M-branes (arXiv:1805.05987)

  (rational equivariant cohomotopy)

• Hisham Sati, Urs Schreiber, Equivariant Cohomotopy implies orientifold tadpole cancellation (arXiv:1909.12277)

  (implying RR-field tadpole cancellation)