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In mathematics, the Thom space, Thom complex, or Pontryagin-Thom construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space. ==Construction of the Thom space== One way to construct this space is as follows. Let :''p'' : ''E'' →''B'' be a rank ''n'' real vector bundle over the paracompact space ''B''. Then for each point ''b'' in ''B'', the fiber ''E''''b'' is a ''n''-dimensional real vector space. We can form an ''n''-sphere bundle ''Sph''(''E'') → ''B'' by taking the one-point compactification of each fiber and gluing them together to get the total space. Finally, from the total space ''Sph''(''E'') we obtain the Thom space ''T''(''E'') as the quotient of ''Sph''(''E'') by ''B''; that is, by identifying all the new points to a single point , which we take as the basepoint of ''T''(''E''). If ''B'' is compact, then ''T''(''E'') is the one-point compactification of ''E''. For example, if ''E'' is the trivial bundle ''B'' × R''n'', then ''Sph''(''E'') is ''B'' × S''n'' and, writing ''B''+ for ''B'' with a disjoint basepoint, ''T''(''E'') is the smash product of ''B''+ and S''n''; that is, the ''n''-th suspension of ''B''+. Alternatively, since ''B'' is paracompact, ''E'' can be given a Euclidean metric and then ''T''(''E'') can be defined as the quotient of the unit disk bundle of ''E'' by the unit (''n''-1)-sphere bundle of ''E''. 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「Thom space」の詳細全文を読む スポンサード リンク
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