nLab stable cohomotopy

Idea

The generalized cohomology theory which is represented by the sphere spectrum is also called stable cohomotopy, as it is the stable homotopy theory version of cohomotopy.

Equivalently, it is the cohomological dual concept to stable homotopy homology theory.

By the Pontryagin-Thom theorem this is equivalently framed cobordism cohomology theory.

Properties

As 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 in: Deitmar 06, p. 2; Guillot 06; Mahanta 17; Dundas, Goodwillie* McCarthy 13, II 1.2; Morava, Connes & Consani 16 and fully explicitly in Chu, Lorscheid & Santhanam 10, Thm. 5.9 and Beardsley & Nakamura 2024, Cor. 2.25. (Chu et al. also generalize to equivariant stable Cohomotopy and equivariant K-theory.)

(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 $A(G) \overset{\text{<a href="https://ncatlab.org/nlab/show/Burnside+ring+is+equivariant+stable+cohomotopy+of+the+point">Segal 71</a>}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Segal-Carlsson+completion+theorem">Carlsson 84</a>}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)$

The third stable framed bordism group

The third stable homotopy group of spheres is the cyclic group of order 24:

$$\array{ \pi_3^s &\simeq& \mathbb{Z}/24 \\ [h_{\mathbb{H}}] &\leftrightarrow& [1] }$$

where the generator $[1] \in \mathbb{Z}/24$ is represented by the quaternionic Hopf fibration $S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4$.

Under the Pontrjagin-Thom isomorphism, identifying the stable homotopy groups of spheres with the bordism ring $\Omega^{fr}_\bullet$ of stably framed manifolds (see at MFr), this generator is represented by the 3-sphere (with its left-invariant framing induced from the identification with the Lie group SU(2) $\simeq$ Sp(1) )

$$\array{ \pi_3^s & \simeq & \Omega_3^{fr} \\ [h_{\mathbb{H}}] & \leftrightarrow & [S^3] \,. }$$

Moreover, the relation $2 4 [S^3] \,\simeq\, 0$ is represented by the complement of 24 open balls inside the K3-manifold (MO:a/44885/381, MO:a/218053/381).

Kahn-Priddy theorem

The Kahn-Priddy theorem characterizes a comparison map between stable cohomotopy and cohomology with coefficients in the infinite real projective space $\mathbb{R}P^\infty \simeq B \mathbb{Z}/2$.

Boardman homomorphisms

To ordinary cohomology

Consider the unit morphism

$$\mathbb{S} \longrightarrow H \mathbb{Z}$$

from the sphere spectrum to the Eilenberg-MacLane spectrum of the integers. For any topological space/spectrum postcomposition with this morphism induces Boardman homomorphisms of cohomology groups (in fact of commutative rings)

(1)$$b^n \;\colon\; \pi^n(X) \longrightarrow H^n(X, \mathbb{Z})$$

from the stable cohomotopy of $X$ in degree $n$ to its ordinary cohomology in degree $n$.

Proposition

(bounds on (co-)kernel of Boardman homomorphism from stable cohomotopy to integral cohomology)

If $X$ is a CW-spectrum which

1. is $(m-1)$-(m-1)-connected

2. of dimension $d \in \mathbb{N}$

then

1. the kernel of the Boardman homomorphism $b^n$ (1) for

   $$m \leq n\leq d -1$$

   is a $\overline{\rho}_{d-n}$-torsion group:

   $$\overline{\rho}_{d-n} ker(b^n) \;\simeq\; 0$$

2. the cokernel of the Boardman homomorphism $b^n$ (1) for

   $$m \leq n \leq d - 2$$

   is a $\overline{\rho}_{d-n-1}$-torsion group:

   $$\overline{\rho}_{d-n-1} coker(b^n) \;\simeq\; 0$$

where

$$\overline{\rho}_{i} \;\coloneqq\; \left\{ \array{ 1 &\vert& i\leq 1 \\ \underoverset{j = 1}{i}{\prod} exp\left( \pi_j\left( \mathbb{S}\right) \right) &\vert& \text{otherwise} } \right.$$

is the product of the exponents of the stable homotopy groups of spheres in positive degree $\leq i$.

(Arlettaz 04, theorem 1.2)

To topological modular forms

Write $\mathbb{S}$ for the sphere spectrum and tmf for the connective spectrum of topological modular forms.

Since tmf is an E-∞ring spectrum, there is an essentially unique homomorphism of E-∞ring spectra

$$\mathbb{S} \overset{e_{tmf}}{\longrightarrow} tmf \,.$$

Regarded as a morphism of generalized homology-theories, this is called the Hurewicz homomorphism, or rather the Boardman homomorphism for $tmf$

Proposition

(Boardman homomorphism in $tmf$ is 6-connected)

The Boardman homomorphism in tmf

$$\mathbb{S} \overset{e_{tmf}}{\longrightarrow} tmf$$

induces an isomorphism on stable homotopy groups (hence from the stable homotopy groups of spheres to the stable homotopy groups of tmf), up to degree 6:

$$\pi_{\bullet \leq 6}(\mathbb{S}) \underoverset{\simeq}{\pi_{\bullet \leq 6}(e_{tmf})}{\longrightarrow} \pi_{\bullet\leq 6}(tmf) \,.$$

(Hopkins 02, Prop. 4.6, DFHH 14, Ch. 13)

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

• Boardman homomorphism

• Spanier-Whitehead duality

flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:

bordism theory$\;$M(B,f) (B-bordism):

• MFr

• MO, MSO, MSpin, MString, …

• MU, MSU, …

  MΩΩSU(n)

  MP, MR

• MSpin^c

• MSp

relative bordism theories:

• MOFr, MUFr, MSUFr

equivariant bordism theory:

• equivariant MFr

• equivariant MO

• equivariant MU

global equivariant bordism theory:

• global equivariant mO,

• global equivariant mU

algebraic:

• algebraic cobordism

References

The concept of stable Cohomotopy as such:

• Frank Adams, part III, section 6, p. 204 of: Stable homotopy and generalised homology, 1974

• John Rognes, p. 1 of: The sphere spectrum, 2004 (pdf)

Discussion of stable Cohomotopy as framed cobordism cohomology theory:

• Robert Stong, Chapter IV, Example 1, p. 40 of Notes on Cobordism theory, Princeton University Press, 1968 (toc pdf, ISBN:9780691649016)

Discussion of stable Cohomotopy of Lie groups:

• C. T. Stretch, Stable cohomotopy and cobordism of abelian groups, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 90, Issue 2 September 1981, pp. 273-278 (doi:10.1017/S0305004100058734)

• Ken-ichi Maruyama, $e$-invariants on the stable cohomotopy groups of Lie groups, Osaka J. Math. Volume 25, Number 3 (1988), 581-589 (euclid:ojm/1200780982)

• Sławomir Nowak, Stable cohomotopy groups of compact spaces, Fundamenta Mathematicae 180 (2003), 99-137 (doi:10.4064/fm180-2-1)

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

• Michael Barratt, Stewart Priddy, On the homology of non-connected monoids and their associated groups, Commentarii Mathematici Helvetici, 47 1 (1972) 1–14 [doi:10.1007/BF02566785, eudml:139496]

• Graeme Segal, Categories and cohomology theories, Topology vol 13, pp. 293-312, 1974 (doi:10.1016/0040-9383(74)90022-6, 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 at AMS Sectional Meeting 1095 (2013) [webpage, pdf]

• Pedro Boavida de Brito, Ieke Moerdijk, Thm. 1.2 in: Dendroidal spaces, $\Gamma$-spaces and the special Barratt-Priddy-Quillen theorem, Journal für die reine und angewandte Mathematik 2020 760 (2020) 229-265 [arXiv:1701.06459, doi:10.1515/crelle-2018-0002]

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) [doi:10.1007/978-1-4471-4393-2]

• 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, 229 4, (2012) 2239-2286 [arxiv:1010.2896, doi:10.1016/j.aim.2011.12.023]

• Jonathan Beardsley, So Nakamura, Projective Geometries and Simple Pointed Matroids as $\mathbb{F}_1$-modules [arXiv:2404.04730]

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, pdf]

The Kahn-Priddy theorem is due to

• Daniel Kahn, Stewart Priddy, Applications of the transfer to stable homotopy theory, Bull. Amer. Math. Soc. Volume 78, Number 6 (1972), 981-987 (Euclid)

Discussion of stable Cohomotopy as framed cobordism cohomology theory:

• Pierre Conner, Edwin Floyd, Section 5 of: The Relation of Cobordism to K-Theories, Lecture Notes in Mathematics 28 Springer 1966 (doi:10.1007/BFb0071091, MR216511)

The relation to β-rings is discussed in

• E. Vallejo, Polynomial operations from Burnside rings to representation functors, J. Pure Appl. Algebra, 65 (1990), pp. 163–190.

• E. Vallejo, Polynomial operations on stable cohomotopy, Manuscripta Math., 67 (1990), pp. 345–365

• E. Vallejo, The free β-ring on one generator, J. Pure Appl. Algebra, 86 (1993), pp. 95–108.

• Guillot 06

see also

• Jack Morava, Rekha Santhanam, Power operations and absolute geometry (pdf)

Discussion of Boardman homomorphisms from stable cohomotopy is in

• Dominique Arlettaz, The generalized Boardman homomorphisms, Central European Journal of Mathematics March 2004, Volume 2, Issue 1, pp 50-56 (doi:10.2478/BF02475949)

A lift of Seiberg-Witten invariants to classes in circle group-equivariant stable cohomotopy is discussed in

• A stable cohomotopy refinement of Seiberg-Witten invariants: I (arXiv:math/0204340)

• A stable cohomotopy refinement of Seiberg-Witten invariants: II (arXiv:math/0204267)

On (stable) motivic Cohomotopy of schemes (as motivic homotopy classes of maps into motivic Tate spheres):

• Aravind Asok, Jean Fasel, Mrinal Kanti Das, Euler class groups and motivic stable cohomotopy, Journal of the EMS 24 8 (2022) 2775–2822 [arXiv:1601.05723, doi:10.4171/jems/1156]

• Samuel Lerbet, Motivic stable cohomotopy and unimodular rows, Advances in Mathematics 436 109415 (2024) [arXiv:2206.11688, doi:10.1016/j.aim.2023.109415]