Cohomotopy groups

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

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 ${\displaystyle \Sigma Y}$, such as a sphere S^q for q${\displaystyle \geq }$1.

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