# nLab stable cohomotopy

Contents

cohomology

### Theorems

#### Spheres

n-sphere

low dimensional n-spheres

# Contents

## 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 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) cohomologyrepresenting
spectrum
equivariant cohomology
of the point $\ast$
cohomology
of classifying space $B G$
(equivariant)
ordinary cohomology
HZBorel equivariance
$H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})$
(equivariant)
complex K-theory
KUrepresentation 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
$R_{\mathbb{F}_p}(G) \simeq K (\mathbb{F}_p)_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {(K \mathbb{F}_p)_G(\ast)} \!\! \overset{\text{Rector 73}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G)$
(equivariant)
stable cohomotopy
$K \mathbb{F}_1 \overset{\text{Segal 74}}{\simeq}$ SBurnside ring
$\mathbb{S}_G(\ast) \simeq A(G)$
Segal-Carlsson completion theorem
$A(G) \overset{\text{Segal 71}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{Carlsson 84}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)$

### The third stable framed bordism group

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

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

where the generator $ \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 $\leq i$.

#### 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)

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

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:

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

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

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

relative bordism theories:

algebraic:

The concept of stable Cohomotopy as such:

Discussion of stable Cohomotopy as framed cobordism cohomology theory:

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

• 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:

The Kahn-Priddy theorem is due to

Discussion of stable Cohomotopy as framed cobordism cohomology theory:

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