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| {{unreferenced|date=October 2013}}
| | Alyson is what my husband enjoys to contact me but I don't like when individuals use my full title. To play lacross is the factor I love most of all. For years she's been living in Kentucky but her spouse wants them to move. Invoicing is what I do.<br><br>Also visit my web blog - authentic psychic readings; [http://Www.Mp3Playa.com/profile.php?u=BeEWRQ mp3playa.com], |
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| In [[mathematics]], particularly [[algebraic topology]], '''cohomotopy sets''' are particular [[category theory|contravariant functors]] from the [[category theory|category]] of pointed [[topological space]]s and point-preserving [[continuous function|continuous]] maps to the category of [[Set (mathematics)|sets]] and [[Function (mathematics)|functions]]. They are [[duality (mathematics)|dual]] to the [[homotopy groups]], but less studied.
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| The ''p''-th cohomotopy set of a pointed [[topological space]] ''X'' is defined by
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| :π<sup> ''p''</sup>(''X'') = [''X'',''S''<sup> ''p''</sup>]
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| the set of pointed [[homotopy]] classes of continuous mappings from ''X'' to the ''p''-[[hypersphere|sphere]] ''S''<sup> ''p''</sup>. For ''p=1'' this set has an [[abelian group]] structure, and is isomorphic to the first [[cohomology]] group ''H<sup>1</sup>(X)'', since ''S''<sup>1</sup> is a [[Eilenberg–MacLane space|''K''('''Z''',1)]]. The set also has a group structure if ''X'' is a [[suspension (topology)|suspension]] <math>\Sigma Y</math>, such as a sphere ''S''<sup>''q''</sup> for ''q''<math>\ge</math>1. | |
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| ==Properties==
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| Some basic facts about cohomotopy sets, some more obvious than others:
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| * π<sup> ''p''</sup>(''S''<sup> ''q''</sup>) = π<sub> ''q''</sub>(''S''<sup> ''p''</sup>) for all ''p'',''q''.
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| * For ''q'' = ''p'' + 1 or ''p'' + 2 ≥ 4, π<sup> ''p''</sup>(''S''<sup> ''q''</sup>) = [[Cyclic group|'''Z'''<sub>2</sub>]]. (To prove this result, [[Lev Pontryagin|Pontrjagin]] developed the concept of framed [[cobordism]]s.)
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| * If ''f'',''g'': ''X'' → ''S''<sup> ''p''</sup> has ||''f''(''x'') - ''g''(''x'')|| < 2 for all ''x'', [''f''] = [''g''], and the homotopy is smooth if ''f'' and ''g'' are.
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| * For ''X'' a compact smooth manifold, π<sup> ''p''</sup>(''X'') is isomorphic to the set of homotopy classes of [[smooth function|smooth]] maps ''X'' → ''S''<sup> ''p''</sup>; in this case, every continuous map can be uniformly approximated by a smooth map and any homotopic smooth maps will be smoothly homotopic.
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| * If ''X'' is an ''m''-[[manifold]], π<sup> ''p''</sup>(''X'') = 0 for ''p'' > ''m''.
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| * If ''X'' is an ''m''-[[manifold]] with boundary, π<sup> ''p''</sup>(''X'',∂''X'') is [[natural isomorphism|canonically]] in [[bijection]] with the set of cobordism classes of [[codimension]]-''p'' framed submanifolds of the [[Interior (topology)|interior]] ''X''-∂''X''.
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| * The [[stable cohomotopy group]] of ''X'' is the [[colimit]]
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| :<math>\pi^p_s(X) = \varinjlim_k{[\Sigma^k X, S^{p+k}]}</math>
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| :which is an [[abelian group]].
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| {{DEFAULTSORT:Cohomotopy Group}}
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| [[Category:Homotopy theory]]
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| {{topology-stub}}
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Alyson is what my husband enjoys to contact me but I don't like when individuals use my full title. To play lacross is the factor I love most of all. For years she's been living in Kentucky but her spouse wants them to move. Invoicing is what I do.
Also visit my web blog - authentic psychic readings; mp3playa.com,