nLab
cohomotopy

Context

Homotopy theory

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).

References

Revised on January 11, 2016 07:36:14 by Urs Schreiber (89.204.155.44)