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 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 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 . (To prove this result, Lev Pontryagin developed the concept of framed cobordism.)
• 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

History

Cohomotopy sets were introduced by Karol Borsuk in 1936. A systematic examination was given by Edwin Spanier in 1949. The stable cohomotopy groups were defined by Franklin P. Peterson in 1956.

References