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Concept
Cohomotopy set
Summary
In mathematics, particularly algebraic topology, cohomotopy sets are particular from the 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 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.
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
which is an abelian group.
Official source
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