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

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

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

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

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

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

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

Properties

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

  • π p ( S q ) = π q ( S p ) {\displaystyle \pi ^{p}(S^{q})=\pi _{q}(S^{p})} for all p and q.
  • For q = p + 1 {\displaystyle q=p+1} and p > 2 {\displaystyle p>2}, the group π p ( S q ) {\displaystyle \pi ^{p}(S^{q})} is equal to Z 2 {\displaystyle \mathbb {Z} _{2}}. (To prove this result, Lev Pontryagin developed the concept of framed cobordism.)
  • If f , g : X → S p {\displaystyle f,g\colon X\to S^{p}} has ‖ f ( x ) − g ( x ) ‖ < 2 {\displaystyle \|f(x)-g(x)\|<2} for all x, then [ f ] = [ g ] {\displaystyle [f]=[g]}, and the homotopy is smooth if f and g are.
  • For X {\displaystyle X} a compact smooth manifold, π p ( X ) {\displaystyle \pi ^{p}(X)} is isomorphic to the set of homotopy classes of smooth maps X → S p {\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 X {\displaystyle X} is an m {\displaystyle m}-manifold, then π p ( X ) = 0 {\displaystyle \pi ^{p}(X)=0} for p > m {\displaystyle p>m}.
  • If X {\displaystyle X} is an m {\displaystyle m}-manifold with boundary, the set π p ( X , ∂ X ) {\displaystyle \pi ^{p}(X,\partial X)} is canonically in bijection with the set of cobordism classes of codimension-p framed submanifolds of the interior X ∖ ∂ X {\displaystyle X\setminus \partial X}.
  • The stable cohomotopy group of X {\displaystyle X} is the colimit

π s p ( X ) = lim → k ⁡ [ Σ k X , S p + k ] {\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. A systematic examination was given by Edwin Spanier in 1949. The stable cohomotopy groups were defined by Franklin P. Peterson in 1956.