Editing Cohomotopy set (section)

==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.