# Cohomotopy group

In mathematics, particularly algebraic topology, **cohomotopy sets** are particular contravariant functors from the category of pointed topological spaces and point-preserving continuous maps to the category of sets and functions. They are dual to the homotopy groups, but less studied.

The *p*-th cohomotopy set of a pointed topological space *X* is defined by

- π
^{ p}(*X*) = [*X*,*S*^{ p}]

the set of pointed homotopy classes of continuous mappings from *X* to the *p*-sphere *S*^{ p}. For *p=1* this set has an abelian group structure, and, provided *X* is a CW-complex, is isomorphic to the first cohomology group *H ^{1}(X)*, since

*S*

^{1}is a

*K*(

**Z**,1). In fact, it is a theorem of Hopf that if

*X*is a CW-complex of dimension at most

*n*, then [

*X*,

*S*

^{ p}] is in bijection with the

*p*-th cohomology group

*H*.

^{ p}(X)The set also has a group structure if *X* is a suspension , such as a sphere *S*^{q} for *q*1.

If *X* is not a CW-complex, *H ^{ 1}(X)* might not be isomorphic to [

*X*,

*S*

^{ 1}]. A counterexample is given by the Warsaw circle, whose first cohomology group vanishes, but admits a map to

*S*

^{1}which is not homotopic to a constant map

^{[1]}

## Properties

Some basic facts about cohomotopy sets, some more obvious than others:

- π
^{ p}(*S*^{ q}) = π_{ q}(*S*^{ p}) for all*p*,*q*. - For
*q*=*p*+ 1 or*p*+ 2 ≥ 4, π^{ p}(*S*^{ q}) =**Z**_{2}. (To prove this result, Pontrjagin developed the concept of framed cobordisms.) - If
*f*,*g*:*X*→*S*^{ p}has ||*f*(*x*) -*g*(*x*)|| < 2 for all*x*, [*f*] = [*g*], and the homotopy is smooth if*f*and*g*are. - For
*X*a compact smooth manifold, π^{ p}(*X*) is isomorphic to the set of homotopy classes of smooth maps*X*→*S*^{ p}; in this case, every continuous map can be uniformly approximated by a smooth map and any homotopic smooth maps will be smoothly homotopic. - If
*X*is an*m*-manifold, π^{ p}(*X*) = 0 for*p*>*m*. - If
*X*is an*m*-manifold with boundary, π^{ p}(*X*,∂*X*) is canonically in bijection with the set of cobordism classes of codimension-*p*framed submanifolds of the interior*X*-∂*X*. - The stable cohomotopy group of
*X*is the colimit

- which is an abelian group.

## References

- ↑ Polish Circle Retrieved July 17, 2014