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



Paths and cylinders

Homotopy groups

Basic facts




Special and general types

Special notions


Extra structure



Manifolds and cobordisms



Where homotopy groups are groups of homotopy classes of maps out spheres, π n(X)[S nX]\pi_n(X)\coloneqq [S^n \to X], cohomotopy groups are groups of homotopy classes into spheres, π n(X)[XS n]\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 AA (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 AA is taken to be a sphere.


Hopf degree theorem


(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).

Relation to cobordism classes of normally framed submanifolds

For XX a closed smooth manifold of dimension DD, the 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.

This construction generalizes to equivariant cohomotopy, see there.

Relation to configuration spaces


(configuration spaces and twisted cohomotopy)

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

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

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


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} \,.


  • 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)

cohomologyequivariant cohomology
non-abelian cohomologycohomotopyequivariant cohomotopy
twisted cohomologytwisted cohomotopy
stable cohomologystable cohomotopyequivariant stable cohomotopy


(equivariant) cohomologyrepresenting
equivariant cohomology
of the point *\ast
of classifying space BGB G
ordinary cohomology
HZBorel equivariance
H G (*)H (BG,)H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})
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)
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)
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="">Rector 73</a>}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G)
stable cohomotopy
K 𝔽 1Segal 74\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="">Segal 71</a>}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{<a href="">Carlsson 84</a>}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)



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)

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

Further discussion includes

See also

Equivariant Cohomotopy

Discussion of the equivariant cohomology-version of cohomotopy (equivariant cohomotopy):

  • Arthur Wasserman, section 3 of Equivariant differential topology, Topology Vol. 8, pp. 127-150, 1969 (pdf)

  • James Cruickshank, Twisted homotopy theory and the geometric equivariant 1-stem, Topology and its Applications Volume 129, Issue 3, 1 April 2003, Pages 251-271 (arXiv:10.1016/S0166-8641(02)00183-9)

  • Daniel Grady, Cobordisms of global quotient orbifolds and an equivariant Pontrjagin-Thom construction (arXiv:1811.08794)

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

and in terms of twisted cohomotopy:

Last revised on April 19, 2019 at 17:13:14. See the history of this page for a list of all contributions to it.