Cohomotopy set
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.
Overview
The p-th cohomotopy set of a pointed topological space X is defined bythe set of pointed homotopy classes of continuous mappings from to the p-sphere.
For p = 1 this set has an abelian group structure, and is called the Bruschlinsky group. Provided is a CW-complex, it is isomorphic to the first cohomology group, since the circle is an Eilenberg–MacLane space of type.
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.
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:- for all p and q.
- For and, the group is equal to.
- If has for all x, then, and the homotopy is smooth if f and g are.
- For a compact smooth manifold, is isomorphic to the set of homotopy classes of smooth maps ; in this case, every continuous map can be uniformly approximated by a smooth map and any homotopic smooth maps will be smoothly homotopic.
- If is an -manifold, then for.
- If is an -manifold with boundary, the set is canonically in bijection with the set of cobordism classes of codimension-p framed submanifolds of the interior.
- The stable cohomotopy group of is the colimit