User:John Z/drafts/cohomotopy groups

From Wikipedia, the free encyclopedia

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

The pth cohomotopy set of a topological space X,

π ^p(X) = [X,S ^p]

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

As the only spheres that are H-spaces are S0, S1, S3, and S7, this set has a natural multiplication only in the cases p = 0, 1, 3, or 7, deriving from real, complex, quaternionic and octonionic multiplication respectively, so we do not get (abelian) groups as in the case of homotopy groups. (The obvious coH-space structure on spheres induces the multiplication in the homotopy groups )

The pth stable cohomotopy group of a space X,

π ^p(X) = :.

π ^p(X) = :[S^kX,S^p+k]

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

• π ^p(S ^q) = π _q(S ^p) for all p,q. (Where we are of course taking either the stable or unstable functors)

• As 'S ^{1 is an Eilenberg-Mac Lane space, the first cohomotopy group is naturally isomorphic to the first cohomology group.}

• ^{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 a compact smooth manifold , 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.