Cohomotopy group

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 [1]


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

which is an abelian group.


  1. Polish Circle Retrieved July 17, 2014

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