Contents

cohomology

# Contents

## Idea

Where homotopy groups are groups of homotopy classes of maps out spheres, $\pi_n(X)\coloneqq [S^n \to X]$, cohomotopy groups are groups of homotopy classes into spheres, $\pi^n(X) \coloneqq [X \to S^n]$.

If instead one considers mapping into the stabilization of the spheres, hence into (some suspension of) the sphere spectrum, then one speaks of stable cohomotopy. In other words, the generalized (Eilenberg-Steenrod) cohomology theory which is represented by the sphere spectrum is stable cohomotopy.

In this vein, regarding terminology: the concept of cohomology (as discussed there) in the very general sense of non-abelian cohomology, is about homotopy classes of maps into any object $A$ (in some (∞,1)-topos). In this way, general non-abelian cohomology is sort of dual to homotopy, and hence might generally be called co-homotopy. This is the statement of Eckmann-Hilton duality. The duality between homotopy (groups) and co-homotopy proper may then be thought of as being the special case of this where $A$ is taken to be a sphere.

## Properties

### Hopf degree theorem

###### Proposition

(Hopf degree theorem)

Let $n \in \mathbb{N}$ be a natural number and $X \in Mfd$ be a connected orientable closed manifold of dimension $n$. Then the $n$th cohomotopy classes $\left[X \overset{c}{\to} S^n\right] \in \pi^n(X)$ of $X$ are in bijection to the degree $deg(c) \in \mathbb{Z}$ of the representing functions, hence the canonical function

$\pi^n(X) \underoverset{\simeq}{S^n \to K(\mathbb{Z},n)}{\longrightarrow} H^n(X,\mathbb{Z}) \;\simeq\; \mathbb{Z}$

from $n$th cohomotopy to $n$th integral cohomology is a bijection.

(e.g. Kosinski 93, IX (5.8))

### Relation to Freudenthal suspension theorem

relation to the Freudenthal suspension theorem (Spanier 49, section 9)

### Smooth representatives

For $X$ a compact smooth manifold, there is a smooth function $X \to S^n$ representing every cohomotopy class (with respect to the standard smooth structure on the sphere manifold).

### Relation to cobordism classes of normally framed submanifolds

For $X$ a closed smooth manifold of dimension $D$, the Pontryagin-Thom construction (e.g. Kosinski 93, IX.5) identifies the set

$SubMfd_{/bord}^{d}(X)$

of cobordism classes of closed and normally framed submanifolds $\Sigma \overset{\iota}{\hookrightarrow} X$ of dimension $d$ inside $X$ with the cohomotopy $\pi^{D-d}(X)$ of $X$ in degree $D- d$

$SubMfd_{/bord}^{d}(X) \underoverset{\simeq}{PT}{\longrightarrow} \pi^{D-d}(X) \,.$

In particular, by this bijection the canonical group structure on cobordism groups in sufficiently high codimension (essentially given by disjoint union of submanifolds) this way induces a group structure on the cohomotopy sets in sufficiently high degree.

This construction generalizes to equivariant cohomotopy, see there.

### Relation to configuration spaces

###### Remark

(configuration spaces and twisted cohomotopy)

The scanning map equivalence (this Prop.) identifies the configuration space of points in $X$ with labels in an n-sphere with the cocycle-space/-infinity-groupoid of $\tau_X$-twisted cohomotopy in degree $\tau + n$, where $\tau_X \coloneqq [S_X(T X)]$ is the class of the spherical fibration of the tangent bundle.

In particular if $X$ is a parallelizable manifold/framed manifold, then $\tau_X = dim(X)$ and the equivalence identifies the configuration space with the plain cohomotopy of $X$ in degree $dim(X) + n$:

$Conf(X,S^n) \;\simeq\; Maps( X, S^{dim(X) + n} ) \,.$

## Examples

### Of 4-Manifolds

Let $X$ be a 4-manifold which is connected and oriented.

The Pontryagin-Thom construction as above gives for $n \in \mathbb{Z}$ the commuting diagram of sets

$\array{ \pi^n(X) &\overset{\simeq}{\longrightarrow}& \mathbb{F}_{4-n}(X) \\ {}^{ \mathllap{h^n} } \downarrow && \downarrow^{ h_{4-n} } \\ H^n(X,\mathbb{Z}) &\underset{\simeq}{\longrightarrow}& H_{4-n}(X,\mathbb{Z}) \,, }$

where $\pi^\bullet$ denotes cohomotopy sets, $H^\bullet$ denotes ordinary cohomology, $H_\bullet$ denotes ordinary homology and $\mathbb{F}_\bullet$ is normally framed cobordism classes of normally framed submanifolds. Finally $h^n$ is the operation of pullback of the generating integral cohomology class on $S^n$ (by the nature of Eilenberg-MacLane spaces):

$h^n(\alpha) \;\colon\; X \overset{\alpha}{\longrightarrow} S^n \overset{generator}{\longrightarrow} B^n \mathbb{Z} \,.$

Now

• $h^0$, $h^1$, $h^4$ are isomorphisms

• $h^3$ is an isomorphism if $X$ is “odd” in that it contains at least one closed oriented surface of odd self-intersection, otherwise $h^3$ becomes an isomorphism on a $\mathbb{Z}/2$-quotient group of $\pi^3(X)$ (which is a group via the group-structure of the 3-sphere (SU(2)))

$\,$

(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
$A(G) \overset{\text{Segal 71}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{Carlsson 84}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)$

## References

### General

Original articles include

• K. Borsuk, Sur les groupes des classes de transformations continues, CR Acad. Sci. Paris 202.1400-1403 (1936): 2 (doi:10.24033/asens.603)

• Edwin Spanier, Borsuk’s Cohomotopy Groups, Annals of Mathematics Second Series, Vol. 50, No. 1 (Jan., 1949), pp. 203-245 (jstor:1969362)

• Franklin P. Peterson, Some Results on Cohomotopy Groups, American Journal of Mathematics Vol. 78, No. 2 (Apr., 1956), pp. 243-258 (jstor:2372514)

• J. Jaworowski, Generalized cohomotopy groups as limit groups, Fundamenta Mathematicae 50 (1962), 393-402 (doi:10.4064/fm-50-4-333-340, pdf)

The relation between cohomotopy classes of manifolds to the cobordism group is discussed for instance in

Further discussion includes