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.
OverviewEdit
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-sphere <math>S^p</math>.<ref>Template:Eom</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 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 structure if <math>X</math> is a 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>"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"</ref>
PropertiesEdit
Template:More citations needed section 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 smooth manifold, <math>\pi^p(X)</math> is isomorphic to the set of homotopy classes of 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 with boundary, the set <math>\pi^p(X,\partial X)</math> is canonically in bijection with the set of cobordism classes of codimension-p framed submanifolds of the 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.
HistoryEdit
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>