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.

Overview

The p-th cohomotopy set of a pointed topological space X is defined by
the set of pointed homotopy classes of continuous mappings from to the p-sphere. For p=1 this set has an abelian group structure, and, provided is a CW-complex, is isomorphic to the first cohomology group, since the circle is an Eilenberg–MacLane space of type. In fact, it is a theorem of Heinz Hopf that if is a CW-complex of dimension at most p, then is in bijection with the p-th cohomology group.
The set also has a natural group structure if is a suspension, such as a sphere for.
If X is not homotopy equivalent to a CW-complex, then might not be isomorphic to. A counterexample is given by the Warsaw circle, whose first cohomology group vanishes, but admits a map to which is not homotopic to a constant map

Properties

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