Puneet Varma (Editor)

Cohomotopy group

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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.

Contents

Overview

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 H1(X), since S1 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 Σ Y , such as a sphere Sq 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 S1 which is not homotopic to a constant map

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) = 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 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
    • π s p ( X ) = lim → k ⁡ [ Σ k X , S p + k ] which is an abelian group.

    References

    Cohomotopy group Wikipedia
    (Text) CC BY-SA