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

π^{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 $K (ℤ, 1)$ .

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 $Σ Y$ , 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:

$π^{p} (S^{q}) = π_{q} (S^{p})$ for all p and q.
For $q = p + 1$ and $p > 2$ , the group $π^{p} (S^{q})$ is equal to $ℤ_{2}$ . (To prove this result, Lev Pontryagin developed the concept of framed cobordism.)
If $f, g : X \to S^{p}$ 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, $π^{p} (X)$ 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 $π^{p} (X) = 0$ for $p > m$ .
If $X$ is an $m$ -manifold with boundary, the set $π^{p} (X, \partial X)$ is canonically in bijection with the set of cobordism classes of codimension-p framed submanifolds of the interior $X ∖ \partial X$ .
The stable cohomotopy group of $X$ is the colimit

π_{s}^{p} (X) = \underset{k}{\lim_{\to}} [Σ^{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.

References

↑ "Cohomotopy_group", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
↑ "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"
↑ K. Borsuk, Sur les groupes des classes de transformations continues, Comptes Rendue de Academie de Science. Paris 202 (1936), no. 1400-1403, 2
↑ 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
↑ F.P. Peterson, Generalized cohomotopy groups, American Journal of Mathematics 78 (1956), 259–281. MR 0084136

[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