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

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

the set of pointed homotopy classes of continuous mappings from ${\displaystyle X}$ to the p-sphere ${\displaystyle S^{p}}$.^[1]

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

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

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

If X is not homotopy equivalent to a CW-complex, then ${\displaystyle H^{1}(X)}$ might not be isomorphic to ${\displaystyle [X,S^{1}]}$. A counterexample is given by the Warsaw circle, whose first cohomology group vanishes, but admits a map to ${\displaystyle S^{1}}$ which is not homotopic to a constant map.^[2]

Properties

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

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

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

which is an abelian group.

History

Cohomotopy sets were introduced by Karol Borsuk in 1936.^[3] A systematic examination was given by Edwin Spanier in 1949.^[4] The stable cohomotopy groups were defined by Franklin P. Peterson in 1956.^[5]