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) = ,S^p/math>

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 $K(\mathbb,1)$. In fact, it is a theorem of Heinz Hopf that if $X$ is a CW-complex of dimension at most ''p'', then ,S^p/math> is in bijection with the ''p''-th cohomology group $H^p(X)$.

The set ,S^p/math> also has a natural group structure if $X$ is a suspension $\Sigma Y$, 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 ,S^1/math>. 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.Polish Circle
Retrieved July 17, 2014.

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 $q= p + 1$ and $p > 2$, the group $\pi^p(S^q)$ is equal to $\mathbb_2$. (To prove this result, Lev Pontryagin developed the concept of framed cobordism.)
* If $f,g\colon X \to S^p$ has $\, f(x) - g(x)\, < 2$ for all ''x'', then = /math>, 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
:$\pi^p_s(X) = \varinjlim_k$
:which is an abelian group.

References

Homotopy theory

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