|
In mathematics, particularly algebraic topology, cohomotopy sets are particular contravariant functors from the category of pointed topological spaces and point-preserving continuous maps to the category of sets and functions. They are dual to the homotopy groups, but less studied. The ''p''-th cohomotopy set of a pointed topological space ''X'' is defined by :π ''p''(''X'') = (''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, provided ''X'' is a CW-complex, is isomorphic to the first cohomology group ''H1(X)'', since ''S''1 is a ''K''(Z,1). In fact, it is a theorem of Hopf that if ''X'' is a CW-complex of dimension at most ''n'', then (''p'' ) is in bijection with the ''p''-th cohomology group ''H p(X)''. The set also has a group structure if ''X'' is a suspension , such as a sphere ''S''''q'' for ''q''1. If ''X'' is not a CW-complex, ''H 1(X)'' might not be isomorphic to (''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 〔(Polish Circle ) Retrieved July 17, 2014〕 ==Properties== Some basic facts about cohomotopy sets, some more obvious than others: * π ''p''(''S'' ''q'') = π ''q''(''S'' ''p'') for all ''p'',''q''. * For ''q'' = ''p'' + 1 or ''p'' + 2 ≥ 4, π ''p''(''S'' ''q'') = Z2. (To prove this result, Pontrjagin developed the concept of framed cobordisms.) * If ''f'',''g'': ''X'' → ''S'' ''p'' has ||''f''(''x'') - ''g''(''x'')|| < 2 for all ''x'', () = (), 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'' → ''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, π ''p''(''X'') = 0 for ''p'' > ''m''. * If ''X'' is an ''m''-manifold with boundary, π ''p''(''X'',∂''X'') is canonically in bijection with the set of cobordism classes of codimension-''p'' framed submanifolds of the interior ''X''-∂''X''. * The stable cohomotopy group of ''X'' is the colimit : :which is an abelian group. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「cohomotopy group」の詳細全文を読む スポンサード リンク
|