nLab
cohomotopy

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

Contents

Idea

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.

Properties

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).

Examples

Of 4-Manifolds

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

The Pontryagin-Thom construction 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 bullet\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 is 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)

References

Last revised on March 16, 2018 at 09:40:02. See the history of this page for a list of all contributions to it.