Cohomotopy group

In mathematics, particularly algebraic topology, cohomotopy groups are contravariant functors from the category of topological spaces and continuous maps to the category of groups and group homomorphisms. They are dual to the homotopy groups, but less studied.

The pth cohomotopy group of a topological space X,

π ^p(X) = [X,S ^p]

is the set of homotopy classes of continuous mappings from X to the p-sphere S ^p.

Some basic facts about cohomotopy groups, 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) = Z₂. (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 compact, π ^p(X) is isomorphic to the group of homotopy classes of smooth maps X → S ^p, every continuous map being uniformly approximable by a smooth map and any homotopic smooth maps being smoothly homotopic.

• If X is an m-manifold, π ^p(X) = 0 for p > m.

• If X is an m-manifold, π ^p(X,∂X) is canonically in bijection with the set of cobordism classes of codimension-p framed submanifolds of the interior of M.
  

If p ≥ 1 + m/2, this is an abelian group with union of disjoint such manifolds as composition.

See also: Cohomotopy group, Abelian group, Algebraic topology, Bijection, Category theory, Cobordism, Codimension, Continuous function, Cyclic group, Duality (mathematics)