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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 to the p- sphere . [1]
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. [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. [5]
This article needs additional citations for
verification. (July 2014) |
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 to the p- sphere . [1]
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. [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. [5]