Cohomotopy set
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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.
Overview
[edit]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]
Properties
[edit]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.
History
[edit]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]
References
[edit]- ^ "Cohomotopy_group", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- ^ "The Polish Circle and some of its unusual properties". Math 205B-2012 Lecture Notes, University of California Riverside. Retrieved November 16, 2023. See also the accompanying diagram "Constructions on the Polish Circle"
- ^ K. Borsuk, Sur les groupes des classes de transformations continues, Comptes Rendue de Academie de Science. Paris 202 (1936), no. 1400-1403, 2
- ^ E. Spanier, Borsuk’s cohomotopy groups, Annals of Mathematics. Second Series 50 (1949), 203–245. MR 29170 https://doi.org/10.2307/1969362 https://www.jstor.org/stable/1969362
- ^ F.P. Peterson, Generalized cohomotopy groups, American Journal of Mathematics 78 (1956), 259–281. MR 0084136