Cohomotopy group

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

The p-th cohomotopy set of a pointed topological space X is defined by

[math]\displaystyle{ \pi^p(X) = [X,S^p] }[/math]

the set of pointed homotopy classes of continuous mappings from [math]\displaystyle{ X }[/math] to the p-sphere [math]\displaystyle{ S^p }[/math]. For p = 1 this set has an abelian group structure, and, provided [math]\displaystyle{ X }[/math] is a CW-complex, is isomorphic to the first cohomology group [math]\displaystyle{ H^1(X) }[/math], since the circle [math]\displaystyle{ S^1 }[/math] is an Eilenberg–MacLane space of type [math]\displaystyle{ K(\mathbb{Z},1) }[/math]. In fact, it is a theorem of Heinz Hopf that if [math]\displaystyle{ X }[/math] is a CW-complex of dimension at most p, then [math]\displaystyle{ [X,S^p] }[/math] is in bijection with the p-th cohomology group [math]\displaystyle{ H^p(X) }[/math].

The set [math]\displaystyle{ [X,S^p] }[/math] also has a natural group structure if [math]\displaystyle{ X }[/math] is a suspension [math]\displaystyle{ \Sigma Y }[/math], such as a sphere [math]\displaystyle{ S^q }[/math] for [math]\displaystyle{ q \ge 1 }[/math].

If X is not homotopy equivalent to a CW-complex, then [math]\displaystyle{ H^1(X) }[/math] might not be isomorphic to [math]\displaystyle{ [X,S^1] }[/math]. A counterexample is given by the Warsaw circle, whose first cohomology group vanishes, but admits a map to [math]\displaystyle{ S^1 }[/math] which is not homotopic to a constant map.

Properties

Some basic facts about cohomotopy sets, some more obvious than others:

• [math]\displaystyle{ \pi^p(S^q) = \pi_q(S^p) }[/math] for all p and q.
• For [math]\displaystyle{ q= p + 1 }[/math] or [math]\displaystyle{ p +2 \ge 4 }[/math], the group [math]\displaystyle{ \pi^p(S^q) }[/math] is equal to [math]\displaystyle{ \mathbb{Z}_2 }[/math]. (To prove this result, Lev Pontryagin developed the concept of framed cobordism.)
• If [math]\displaystyle{ f,g\colon X \to S^p }[/math] has [math]\displaystyle{ \|f(x) - g(x)\| \lt 2 }[/math] for all x, then [math]\displaystyle{ [f] = [g] }[/math], and the homotopy is smooth if f and g are.
• For [math]\displaystyle{ X }[/math] a compact smooth manifold, [math]\displaystyle{ \pi^p(X) }[/math] is isomorphic to the set of homotopy classes of smooth maps [math]\displaystyle{ 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]\displaystyle{ X }[/math] is an [math]\displaystyle{ m }[/math]-manifold, then [math]\displaystyle{ \pi^p(X)=0 }[/math] for [math]\displaystyle{ p \gt m }[/math].
• If [math]\displaystyle{ X }[/math] is an [math]\displaystyle{ m }[/math]-manifold with boundary, the set [math]\displaystyle{ \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]\displaystyle{ X \setminus \partial X }[/math].
• The stable cohomotopy group of [math]\displaystyle{ X }[/math] is the colimit

[math]\displaystyle{ \pi^p_s(X) = \varinjlim_k{[\Sigma^k X, S^{p+k}]} }[/math]

which is an abelian group.