In mathematics, particularly algebraic topology, cohomotopy sets are particular contravariant functors from the category of pointed topological spaces and basepoint-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
[1] For p = 1 this set has an abelian group structure, and is called the Bruschlinsky group.
is a CW-complex, it is isomorphic to the first cohomology group
A theorem of Heinz Hopf states that if
is a CW-complex of dimension at most p, then
is in bijection with the p-th cohomology group
also has a natural group structure if
If X is not homotopy equivalent to a CW-complex, then
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.
[2] Some basic facts about cohomotopy sets, some more obvious than others: Cohomotopy sets were introduced by Karol Borsuk in 1936.
[3] A systematic examination was given by Edwin Spanier in 1949.
[4] The stable cohomotopy groups were defined by Franklin P. Peterson in 1956.