Editing Cohomotopy set

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In [[mathematics]], particularly [[algebraic topology]], '''cohomotopy sets''' are particular [[category theory|contravariant functors]] from the [[category (mathematics)|category]] of [[pointed space|pointed topological spaces]] and basepoint-preserving [[continuous function (topology)|continuous]] maps to the category of [[Set (mathematics)|sets]] and [[Function (mathematics)|functions]]. They are [[duality (mathematics)|dual]] to the [[homotopy groups]], but less studied.

==Overview==
The ''p''-th cohomotopy set of a pointed [[topological space]] ''X'' is defined by

:<math>\pi^p(X) = [X,S^p]</math>

the set of pointed [[homotopy]] classes of continuous mappings from <math>X</math> to the ''p''-[[hypersphere|sphere]] <math>S^p</math>.<ref>{{eom|title=Cohomotopy_group}}</ref>

For ''p'' = 1 this set has an [[abelian group]] structure, and is called the '''Bruschlinsky group'''. Provided <math>X</math> is a [[CW-complex]], it is [[group isomorphism|isomorphic]] to the first [[cohomology]] group <math>H^1(X)</math>, since the [[circle]] <math>S^1</math> is an [[Eilenberg–MacLane space]] of type <math>K(\mathbb{Z},1)</math>. 

A theorem of [[Heinz Hopf]] states that if <math>X</math> is a [[CW-complex]] of dimension at most ''p'', then  <math>[X,S^p]</math> is in [[bijection]] with the ''p''-th cohomology group <math>H^p(X)</math>.

The set <math>[X,S^p]</math> also has a natural [[group (mathematics)|group]] structure if <math>X</math> is a [[suspension (topology)|suspension]] <math>\Sigma Y</math>, such as a sphere <math>S^q</math> for <math>q \ge 1</math>.

If ''X'' is not homotopy equivalent to a CW-complex, then <math>H^1(X)</math> might not be isomorphic to <math>[X,S^1]</math>. A counterexample is given by the [[Warsaw circle]], whose first cohomology group vanishes, but admits a map to <math>S^1</math> which is not homotopic to a constant map.<ref>"[http://math.ucr.edu/~res/math205B-2012/polishcircle.pdf 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 "[http://math.ucr.edu/~res/math205B-2012/polishcircleA.pdf Constructions on the Polish Circle]"</ref>

==Properties==

{{More citations needed section|date=December 2024}}
Some basic facts about cohomotopy sets, some more obvious than others:

* <math>\pi^p(S^q) = \pi_q(S^p)</math> for all ''p'' and ''q''.
* For <math>q= p + 1</math> and <math>p > 2</math>, the group <math>\pi^p(S^q)</math> is equal to <math>\mathbb{Z}_2</math>. (To prove this result, [[Lev Pontryagin]] developed the concept of framed [[cobordism]].)
* If <math>f,g\colon X \to S^p</math> has <math>\|f(x) - g(x)\| < 2</math> for all ''x'', then <math>[f] = [g]</math>, and the homotopy is smooth if ''f'' and ''g'' are.
* For <math>X</math> a [[compact space|compact]] [[smooth manifold]], <math>\pi^p(X)</math> is isomorphic to the set of homotopy classes of [[smooth function|smooth]] maps <math>X \to S^p</math>; in this case, every continuous map can be uniformly approximated by a smooth map and any homotopic smooth maps will be smoothly homotopic.
* If <math>X</math> is an <math>m</math>-[[manifold]], then <math>\pi^p(X)=0</math> for <math>p > m</math>.
* If <math>X</math> is an <math>m</math>-[[manifold#Manifold with boundary|manifold with boundary]], the set <math>\pi^p(X,\partial X)</math> is [[natural isomorphism|canonically]] in bijection with the set of cobordism classes of [[codimension]]-''p'' framed submanifolds of the [[Interior (topology)|interior]] <math>X \setminus \partial X</math>.
* The [[stable cohomotopy group]] of <math>X</math> is the [[colimit]]
:<math>\pi^p_s(X) = \varinjlim_k{[\Sigma^k X, S^{p+k}]}</math>
:which is an abelian group.

==History==
Cohomotopy sets were introduced by [[Karol Borsuk]] in 1936.<ref>K. Borsuk, ''Sur les groupes des classes de transformations continues'', Comptes Rendue de Academie de Science. Paris '''202''' (1936), no. 1400-1403, 2</ref>  A systematic examination was given by [[Edwin Spanier]] in 1949.<ref>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</ref> The stable cohomotopy groups were defined by [[Franklin P. Peterson]] in 1956.<ref>F.P. Peterson, ''Generalized cohomotopy groups'', American Journal of Mathematics '''78''' (1956), 259–281. MR 0084136</ref>

==References==
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[[Category:Homotopy theory]]