nLab
cohomotopy

Contents

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Manifolds and cobordisms

Contents

Idea

General

Cohomotopy cohomology theory π \pi^\bullet is the (non-abelian) generalized cohomology theory whose cocycle spaces are spaces of maps into an n-sphere, hence whose cohomology classes are homotopy classes of maps into an n-sphere:

π n(X)Maps(X,S n)/ homotopy \pi^n(X) \;\coloneqq\; Maps\big( X, S^n\big)/_{\sim_{homotopy}}

So, dually to how homotopy groups

π n(X)Maps(S n,X)/ homotopy \pi_n(X)\coloneqq Maps\big( S^n, X\big)/_{\sim_{homotopy}}

are groups of homotopy classes of maps out of spheres, Cohomotopy sets are sets of homotopy classes of maps into spheres, whence the dual name.

If instead one considers only the stable aspect of Cohomotopy sets, by mapping into the stabilization of the spheres, hence into (some suspension of) the sphere spectrum, then one speaks of stable Cohomotopy, written

𝕊 n(X)Maps(Σ X,Σ S n)/ homotopy. \mathbb{S}^n(X) \;\coloneqq\; Maps \big( \Sigma^\infty X , \Sigma^{\infty} S^n \big)/_{\sim_{homotopy}} \,.

In other words, the generalized (Eilenberg-Steenrod) cohomology theory which is represented by the sphere spectrum is stable Cohomotopy.

Remark

(terminology)

Therefore “Cohomotopy theory” is really shorthand for “Cohomotopy cohomology theory” and as such is dual to homotopy homology theory, which in the stable case is known as stable homotopy homology theory.

In particular, cohomotopy theory is a concrete particular and not dual to the abstract general of homotopy theory; and is hence also not on par with the abstract general of cohomology theory. Rather, Cohomotopy theory is one instance of a cohomology theory, and as such is a sibling of ordinary cohomology theory (HR-theory)), K-theory, etc.

To emphasize this, one might, in the stable case, say 𝕊\mathbb{S}-theory instead of “stable Cohomotopy theory”; where 𝕊\mathbb{S} denotes the sphere spectrum. In the unstable case there is no widely adopted notation, but one might consider saying “π\mathbf{\pi}-theory” (with π\pi the established symbol for (co)homotopy groups) or SS-theory (with “SS” for n-spheres S nS^n) for unstable Cohomotopy theory.

In any case, to highlight that Cohomotopy theory is a concrete particular and not an abstract general, it makes good sense to capitalize the term and speak of Cohomotopy cohomology theory or just Cohomotopy theory, for short.

The following table indicates the pattern:

generalized cohomology theorygeneralized homology theoryclassifying spacerepresenting spectrum
Cohomotopy theory
(non-abelian)
n-sphere
S nS^n
stable Cohomotopy theory
(abelian)
stable homotopy homology theorysphere spectrum
𝕊\mathbb{S}
\,
ordinary cohomology:
HR-cohomology theory
ordinary homology: HR-homology theoryEilenberg-MacLane space K(R,n)K(R,n)Eilenberg-MacLane spectrum
HRH R
K-cohomology theoryK-homology theorystable unitary group
BU
K-theory spectrum
KU, …

As for any generalized cohomology theory there are immediate variants to plain Cohomotopy theory, as shown in the following table:

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

As the absolute cohomology theory

In some sense Cohomotopy is the most fundamental of all generalized cohomology theories (“The Music of the Spheres”).

Concretely, stable Cohomotopy cohomology theory is the initial object among multiplicative cohomology theories, in that the sphere spectrum is the initial object in (E-infinity) ring spectra. This means that for any other multiplicative cohomology theory EE there is an essentially unique multiplicative natural transformation

(1)𝕊 (X)β(X)E (X) \mathbb{S}^\bullet(X) \overset{\beta(X)}{\longrightarrow} E^\bullet(X)

from Cohomotopy cohomology groups to EE-cohomology groups – the Boardman homomorphism.

Specifically for E=K𝔽E = K \mathbb{F} the algebraic K-theory of a field 𝔽\mathbb{F} (such as a prime field 𝔽 p\mathbb{F}_p) there is such a comparison morphism; and another way how stable Cohomotopy is the most fundamental of all K-theories is that it is equivalently the algebraic K-theory over the “absolute base”, namely over the “field with one element𝔽 1\mathbb{F}_1 (see there for more):

𝕊 (X)K𝔽 1 (X). \mathbb{S}^\bullet(X) \;\simeq\; K\mathbb{F}_1 ^ \bullet(X) \,.

For example, for E=E = KU and in the case of GG-equivariant cohomology theory (equivariant Cohomotopy theory and equivariant K-theory) the Boardman homomorphism (1) gives the comparison map

𝕊 G 0(*)R 𝔽 1(G)A(G)[]R (G)KU G 0(*) \mathbb{S}^0_G(\ast) \simeq R_{\mathbb{F}_1}(G) \simeq A(G) \overset{\mathbb{C}[-]}{\longrightarrow} R_{\mathbb{C}}(G) \simeq KU^0_G(\ast)

from the Burnside ring to the representation ring of the finite group GG, by forming permutation representations; where we may think of the Burnside ring as being the representation ring over the “field with one element” (see e.g. Chu-Lorscheid-Santhanam 10), as indicated above.

So far, this applies to stable Cohomotopy theory, which historically has received almost all the attention. But, while stabilization makes the immensely rich nature of homotopy theory a tad more tractable, it is only an approximation (just the first Goodwillie derivative!) of full unstable/non-abelian cohomology. Hence the one concept more fundamental than stable Cohomotopy theory is actual Cohomotopy theory.

For example, the classification of Yang-Mills instantons on 4\mathbb{R}^4 is typically regarded in the non-abelian cohomology theory represented by the classifying space BSU(N)B SU(N) of the special unitary group (for N2N \geq 2, starting with SU(2))

(BSU(N)) 0(( 4) cpt){( 4) cptBSU(N)}/ homotopyinstantonnumber. (B SU(N))^0 \Big( \big( \mathbb{R}^{4} \big)^{cpt} \Big) \;\coloneqq\; \Big\{ \big( \mathbb{R}^{4} \big)^{cpt} \to B SU(N) \Big\}/_{\sim_{homotopy}} \;\simeq\; \underset{ \mathclap{ \color{blue} {\text{instanton} \atop \text{number}} } }{ \mathbb{Z} } \,.

But since the one-point compactification of 4d Euclidean space is the 4-sphere ( 4) cptS 4\big( \mathbb{R}^4\big)^{cpt} \simeq S^4, this classification factors through one in unstable Cohomotopy theory, via the “unstable Boardman homomorphism” S 4BSU(N)S^4 \longrightarrow B SU(N) representing the generator of the 4th homotopy group of BSU(N)B SU(N) (see there)

π 0(( 4) cpt){( 4) cptS 4}/ homotopyinstantonnumber. \pi^0 \Big( \big( \mathbb{R}^{4} \big)^{cpt} \Big) \;\coloneqq\; \Big\{ \big( \mathbb{R}^{4} \big)^{cpt} \to S^4 \Big\}/_{\sim_{homotopy}} \;\simeq\; \underset{ \mathclap{ \color{blue} {\text{instanton} \atop \text{number}} } }{ \mathbb{Z} } \,.

(see SS 19, p. 9-10) This is the tip of an iceberg. Which needs to be discussed elsewhere.


Properties

Hopf degree theorem

Proposition

(Hopf degree theorem)

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

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

from nnth cohomotopy to nnth integral cohomology is a bijection.

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

Poincaré–Hopf theorem

Relation to Freudenthal suspension theorem

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

Smooth representatives

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

PT-Construction and normally framed submanifolds

For XX a closed smooth manifold of dimension DD, the assignment of Cohomotopy charge (Pontryagin-Thom construction, e.g. Kosinski 93, IX.5) identifies the set

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

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

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

(e.g. Kosinski 93, IX Theorem (5.5))

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.

\,

graphics grabbed form Sati-Schreiber 19

Here the normal framing of the submanifolds plays the role of the charge in Cohomotopy which they carry:

\,

graphics grabbed form Sati-Schreiber 19

For example:

\,

graphics grabbed form Sati-Schreiber 19

This construction generalizes to equivariant cohomotopy, see there.

With the equivariant Hopf degree theorem the above example has the following 2\mathbb{Z}_2-equivariant version (see there):

\,

graphics grabbed form Sati-Schreiber 19

Further by the equivariant Hopf degree theorem (see there), this example generalizes to equivariant cohomotopy of toroidal orientifolds:

\,

graphics grabbed form Sati-Schreiber 19


Cohomotopy charge map and Relation to configuration spaces

The Cohomotopy charge map is the function that assigns to a configuration of points their total charge as measured in Cohomotopy-cohomology theory.

This is alternatively known as the “electric field map” (Salvatore 01 following Segal 73, Section 1, see also Knudsen 18, p. 49) or the “scanning map” (Kallel 98).

For DD \in \mathbb{N} the Cohomotopy charge map is the continuous function

(2)Conf( D)ccπ D(( D) cpt)=Maps */(( D) cpt,S D)=Ω DS D Conf\big( \mathbb{R}^D \big) \overset{cc}{\longrightarrow} \mathbf{\pi}^D \Big( \big( \mathbb{R}^D \big)^{cpt} \Big) = Maps^{\ast/\!}\Big( \big(\mathbb{R}^D\big)^{cpt} , S^D\big) = \Omega^{D} S^D

from the configuration space of points in the Euclidean space D\mathbb{R}^D to the DD-Cohomotopy cocycle space vanishing at infinity on the Euclidean space(which is equivalently the space of pointed maps from the one-point compactification S D( D)S^D \simeq \big( \mathbb{R}^D \big) to itself, and hence equivalently the DD-fold iterated based loop space of the D-sphere), which sends a configuration of points in D\mathbb{R}^D, each regarded as carrying unit charge to their total charge as measured in Cohomotopy-cohomology theory (Segal 73, Section 3).

This has evident generalizations to other manifolds than just Euclidean spaces, to spaces of labeleed configurations and to equivariant Cohomotopy. The following graphics illustrates the Cohomotopy charge map on G-space tori for G= 2G = \mathbb{Z}_2 with values in 2\mathbb{Z}_2-equivariant Cohomotopy:

graphics grabbed from SS 19

In some situations the Cohomotopy charge map is a weak homotopy equivalence and hence exhibits, for all purposes of homotopy theory, the Cohomotopy cocycle space of Cohomotopy charges as an equivalent reflection of the configuration space of points.

Proposition

(group completion on configuration space of points is iterated based loop space)

The Cohomotopy charge map (2)

Conf( D)ccΩ DS D Conf \big( \mathbb{R}^D \big) \overset{ cc }{\longrightarrow} \Omega^D S^D

from the full unordered and unlabeled configuration space (here) of Euclidean space D\mathbb{R}^D to the DD-fold iterated based loop space of the D-sphere, exhibits the group completion (here) of the configuration space monoid

ΩB Conf( D)Ω DS D \Omega B_{{}_{\sqcup}\!} Conf \big( \mathbb{R}^D \big) \overset{ \simeq }{\longrightarrow} \Omega^D S^D

(Segal 73, Theorem 1)

Proposition

(Cohomotopy charge map is weak homotopy equivalence on sphere-labeled configuration space of points)

For D,kD, k \in \mathbb{N} with k1k \geq 1, the Cohomotopy charge map (2)

Conf( D,S k)ccΩ DS D+k Conf \big( \mathbb{R}^D, S^k \big) \underoverset{\simeq}{cc}{\longrightarrow} \Omega^D S^{D + k}

is a weak homotopy equivalence from the configuration space (here) of unordered points with labels in S kS^k and vanishing at the base point of the label space to the DD-fold iterated loop space of the D+k-sphere.

(Segal 73, Theorem 3)

The May-Segal theorem generalizes from Euclidean space to closed smooth manifolds if at the same time one passes from plain Cohomotopy to twisted Cohomotopy, twisted, via the J-homomorphism, by the tangent bundle:

Proposition

Let

  1. X nX^n be a smooth closed manifold of dimension nn;

  2. 1k1 \leq k \in \mathbb{N} a positive natural number.

Then the Cohomotopy charge map constitutes a weak homotopy equivalence

Maps /BO(n)(X n,S n def+k trivO(n))a J-twisted Cohomotopy spaceCohomotopy charge mapConf(X n,S k)aconfiguration spaceof points \underset{ \color{blue} { \phantom{a} \atop \text{ J-twisted Cohomotopy space}} }{ Maps_{{}_{/B O(n)}} \Big( X^n \;,\; S^{ \mathbf{n}_{def} + \mathbf{k}_{\mathrm{triv}} } \!\sslash\! O(n) \Big) } \underoverset {\simeq} { \color{blue} \text{Cohomotopy charge map} } {\longleftarrow} \underset{ \mathclap{ \color{blue} { \phantom{a} \atop { \text{configuration space} \atop \text{of points} } } } }{ Conf \big( X^n, S^k \big) }

between

  1. the J-twisted (n+k)-Cohomotopy space of X nX^n, hence the space of sections of the (n+k)(n + k)-spherical fibration over XX which is associated via the tangent bundle by the O(n)-action on S n+k=S( n× k+1)S^{n+k} = S(\mathbb{R}^{n} \times \mathbb{R}^{k+1})

  2. the configuration space of points on X nX^n with labels in S kS^k.

(Bödigheimer 87, Prop. 2, following McDuff 75)

Remark

In the special case that the closed manifold X nX^n in Prop. is parallelizable, hence that its tangent bundle is trivializable, the statement of Prop. reduces to this:

Let

  1. X nX^n be a parallelizable closed manifold of dimension nn;

  2. 1k1 \leq k \in \mathbb{N} a positive natural number.

Then the Cohomotopy charge map constitutes a weak homotopy equivalence

Maps(X n,S n+k)a Cohomotopy spaceCohomotopy charge mapConf(X n,S k)aconfiguration spaceof points \underset{ \color{blue} { \phantom{a} \atop \text{ Cohomotopy space}} }{ Maps \Big( X^n \;,\; S^{ n + k } \Big) } \underoverset {\simeq} { \color{blue} \text{Cohomotopy charge map} } {\longleftarrow} \underset{ \mathclap{ \color{blue} { \phantom{a} \atop { \text{configuration space} \atop \text{of points} } } } }{ Conf \big( X^n, S^k \big) }

between

  1. (n+k)(n+k)-Cohomotopy space of X nX^n, hence the space of maps from XX to the (n+k)-sphere

  2. the configuration space of points on X nX^n with labels in S kS^k.


Complex-rational Cohomotopy and moduli space of Yang-0Mills monopoles

The assignment of scattering amplitudes of monopoles in SU(2)-Yang-Mills theory is a diffeomorphism

kSR k \mathcal{M}_k \overset{ S }{\longrightarrow} R_k

identifying the moduli space of monopoles of number kk with the space of complex-rational functions form the Riemann sphere to itself, of degree kk (hence the cocycle space of complex-rational 2-Cohomotopy).

(Atiyah-Hitchin 88, Theorem 2.10)

\,

This is a non-abelian analog of the Dirac charge quantization of the electromagnetic field, with ordinary cohomology replaced by Cohomotopy cohomology theory.

Examples

Of 4-Manifolds

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

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

π n(X) 𝔽 4n(X) h n h 4n H n(X,) H 4n(X,), \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 π \pi^\bullet denotes cohomotopy sets, H H^\bullet denotes ordinary cohomology, H H_\bullet denotes ordinary homology and 𝔽 \mathbb{F}_\bullet is normally framed cobordism classes of normally framed submanifolds. Finally h nh^n is the operation of pullback of the generating integral cohomology class on S nS^n (by the nature of Eilenberg-MacLane spaces):

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

Now

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

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

(Kirby-Melvin-Teichner 12)


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


(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)


References

General

Original articles include

See also

The unstable Pontrjagin-Thom theorem identifying cobordism classes of normally framed submanifolds with their Cohomotopy charge is discussed for instance in:

Further discussion includes

Cohomotopy cocycle spaces

Discussion of Cohomotopy cocycle spaces (i.e. spaces of maps into an n-sphere):

Discussion of cocycle spaces for rational Cohomotopy (see also at rational model of mapping spaces):

  • Jesper Møller, Martin Raussen, Rational Homotopy of Spaces of Maps Into Spheres and Complex Projective Spaces, Transactions of the American Mathematical Society Vol. 292, No. 2 (Dec., 1985), pp. 721-732 (jstor:2000242)

Relation to knots and links:

Application to persistency

Application of Cohomotopy similar to that of persistent homology:

Equivariant Cohomotopy

Discussion of the stable cohomotopy (framed cobordism cohomology theory) in the equivariant cohomology-version of cohomotopy (equivariant cohomotopy):

and in the twisted cohomology-version (twisted cohomotopy)

Discussion of M-brane physics in terms of rational equivariant cohomotopy:

and in terms of twisted cohomotopy:

In spectral geometry

Discussion of smooth functions into the 4-sphere in the context of Connes-Lott models in spectral non-commutative geometry:

Last revised on November 15, 2019 at 04:01:56. See the history of this page for a list of all contributions to it.