In

Polish Circle

Retrieved July 17, 2014.

mathematics
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, particularly algebraic topology
250px, A torus, one of the most frequently studied objects in algebraic topology
Algebraic topology is a branch of mathematics
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, cohomotopy sets are particular contravariant functors from the category
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of pointed topological spaces and basepoint-preserving continuous
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maps to the category of sets and functions
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. They are dual
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to the homotopy groups
In mathematics
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, but less studied.
Overview

The ''p''-th cohomotopy set of a pointed topological space ''X'' is defined by :$\backslash pi^p(X)\; =\; [X,S^p]$ the set of pointed homotopy classes of continuous mappings from $X$ to the ''p''-hypersphere, sphere $S^p$. For ''p'' = 1 this set has an abelian group structure, and, provided $X$ is a CW-complex, is group isomorphism, isomorphic to the first cohomology group $H^1(X)$, since the circle $S^1$ is an Eilenbergâ€“MacLane space of type $K(\backslash mathbb,1)$. In fact, it is a theorem of Heinz Hopf 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 (mathematics), group structure if $X$ is a suspension (topology), suspension $\backslash Sigma\; Y$, such as a sphere $S^q$ for $q\; \backslash ge\; 1$. 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.Retrieved July 17, 2014.

Properties

Some basic facts about cohomotopy sets, some more obvious than others: * $\backslash pi^p(S^q)\; =\; \backslash pi\_q(S^p)$ for all ''p'' and ''q''. * For $q=\; p\; +\; 1$ and $p\; >\; 2$, the group $\backslash pi^p(S^q)$ is equal to $\backslash mathbb\_2$. (To prove this result, Lev Pontryagin developed the concept of framed cobordism.) * If $f,g\backslash colon\; X\; \backslash to\; S^p$ has $\backslash ,\; f(x)\; -\; g(x)\backslash ,\; <\; 2$ for all ''x'', then $[f]\; =\; [g]$, and the homotopy is smooth if ''f'' and ''g'' are. * For $X$ a compact space, compact smooth manifold, $\backslash pi^p(X)$ is isomorphic to the set of homotopy classes of smooth function, smooth maps $X\; \backslash to\; S^p$; 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 $\backslash pi^p(X)=0$ for $p\; >\; m$. * If $X$ is an $m$-manifold#Manifold with boundary, manifold with boundary, the set $\backslash pi^p(X,\backslash partial\; X)$ is natural isomorphism, canonically in bijection with the set of cobordism classes of codimension-''p'' framed submanifolds of the Interior (topology), interior $X\; \backslash setminus\; \backslash partial\; X$. * The stable cohomotopy group of $X$ is the colimit :$\backslash pi^p\_s(X)\; =\; \backslash varinjlim\_k$ :which is an abelian group.References

Homotopy theory {{topology-stub