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 to the p-sphere.

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, since the circle is an Eilenberg–MacLane space.

A theorem of Heinz Hopf is a CW-complex of dimension at most p is in bijection with the p.

also has a natural group is a suspension.

If X. A counterexample is given by the Warsaw circle 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.

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

• for all x, and the homotopy is smooth if f and g are.

• a compact smooth manifold is isomorphic to the set of homotopy classes of smooth; in this case, every continuous map can be uniformly approximated by a smooth map and any homotopic smooth maps will be smoothly homotopic.

• -manifold.

• -manifold with boundary is canonically in bijection with the set of cobordism classes of codimension-p framed submanifolds of the interior.

• The stable cohomotopy group is the colimit

• which is an abelian group.

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