cohomology

# Contents

## Idea

Where homotopy groups are groups of homotopy classes of maps out spheres, $\pi_n(X)\coloneqq [S^n \to X]$, cohomotopy groups are groups of homotopy classes into spheres, $\pi^n(X) \coloneqq [X \to S^n]$.

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 $A$ (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 $A$ 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 $X$ a compact smooth manifold, there is a smooth function $X \to S^n$ 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)