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]

References

1. "Cohomotopy_group", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
2. "The Polish Circle and some of its unusual properties". Math 205B-2012 Lecture Notes, University of California Riverside. Retrieved November 16, 2023. See also the accompanying diagram "Constructions on the Polish Circle"
3. K. Borsuk, Sur les groupes des classes de transformations continues, Comptes Rendue de Academie de Science. Paris 202 (1936), no. 1400-1403, 2
4. E. Spanier, Borsuk’s cohomotopy groups, Annals of Mathematics. Second Series 50 (1949), 203–245. MR 29170 https://doi.org/10.2307/1969362 https://www.jstor.org/stable/1969362
5. F.P. Peterson, Generalized cohomotopy groups, American Journal of Mathematics 78 (1956), 259–281. MR 0084136