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.


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

Let XX be a smooth manifold of dimension nn \in \mathbb{N} and let knk \leq n. Then the Pontryagin-Thom construction induces a bijection

[X,S k]Ω nk(X) [X, S^k] \overset{\simeq}{\longrightarrow} \Omega^{n-k}(X)

from the cohomotopy sets of XX to the cobordism group of (nk)(n-k)-dimensional submanifolds with normal framing up to normally framed cobordism.

In particular, the natural group structure on cobordism group (essentially given by disjoint union of submanifolds) this way induces a group structure on the cohomotopy sets.

This is made explicit for instance in Kosinski 93, chapter IX.


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)


Original articles include

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

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

Further discussion includes

See also

Last revised on May 18, 2018 at 06:32:37. See the history of this page for a list of all contributions to it.