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 $q\geq 1$.

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

$\pi _{s}^{p}(X)=\varinjlim _{k}{[\Sigma ^{k}X,S^{p+k}]}$

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.