In mathematics, particularly algebraic topology, cohomotopy sets are particular contravariant functors from the category of pointed topological spaces and basepoint-preserving continuous maps to the category of sets and functions.
They are dual to the homotopy groups, but less studied.
Overview
The p-th cohomotopy set of a pointed topological space X is defined by
\pi^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 the circle S^1 is an Eilenberg–MacLane space of type K(\mathbb{Z},1).
In fact, it is a theorem of Heinz Hopf that if X is a CW-complex of dimension at most p, then  [X,S^p] is in bijection with the p-th cohomology group H^p(X).
The set [X,S^p] also has a natural group structure if X is a suspension \Sigma Y, such as a sphere S^q for q \ge 1.
If X is not homotopy equivalent to a CW-complex, then 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.Polish Circle.
Retrieved July 17, 2014.
Properties
Some basic facts about cohomotopy sets, some more obvious than others:
\pi^p(S^q) = \pi_q(S^p) for all p and q.
For q= p + 1 and p > 2, the group \pi^p(S^q) is equal to \mathbb{Z}_2.
(To prove this result, Lev Pontryagin developed the concept of framed cobordism.)
If f,g\colon X \to S^p has \|f(x) - g(x)\| < 2 for all x, then [f] = [g], and the homotopy is smooth if f and g are.
For X a compact smooth manifold, \pi^p(X) is isomorphic to the set of homotopy classes of smooth maps X \to 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, then \pi^p(X)=0 for p > m.
If X is an m-manifold with boundary, the set \pi^p(X,\partial X) is canonically in bijection with the set of cobordism classes of codimension-p framed submanifolds of the interior X \setminus \partial X.
The stable cohomotopy group of X is the colimit
\pi^p_s(X) = \varinjlim_k{[\Sigma^k X, S^{p+k}]}
which is an abelian group.
References
