Cohomotopy set

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


 * $$\pi^p(X) = [X,S^p]$$

the set of pointed homotopy classes of continuous mappings from $$X$$ to the p-sphere $$S^p$$.

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

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

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

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

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


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

History
Cohomotopy sets were introduced by Karol Borsuk in 1936. A systematic examination was given by Edwin Spanier in 1949. The stable cohomotopy groups were defined by Franklin P. Peterson in 1956.