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

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

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, provided $X$ is a CW-complex, is isomorphic to the first cohomology group $H^1(X)$, since the circle $S^1$ is an Eilenberg–MacLane space of type $K(\mathbb{\mathbb{Z}},1)$. In fact, it is a theorem of Heinz Hopf 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 $\mathrm{\Sigma }Y$, 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 $q=p+1$ or $p+2\ge 4$, the group ${\pi }^p(S^q)$ is equal to ${\mathbb{\mathbb{Z}}}_2$. (To prove this result, Lev Pontryagin developed the concept of framed cobordism.)
• If $f,g:X\to S^p$ has $\Vert f(x)-g(x)\Vert <2$ 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

${\pi }_s^p(X)=\underset{k}{\underrightarrow{\lim }}[{\mathrm{\Sigma }}^{k}X,S^{p+k}]$

which is an abelian group.