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.
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 , such as a sphere Sq for q1.
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
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
- which is an abelian group.
- Polish Circle Retrieved July 17, 2014