Editing Cohomotopy set (section)

==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>