Cohomotopy group

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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; 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, 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.