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In mathematics, the Pontryagin classes, named for Lev Pontryagin, are certain characteristic classes. The Pontryagin class lies in cohomology groups with degree a multiple of four. It applies to real vector bundles. == Definition == Given a real vector bundle ''E'' over ''M'', its ''k''-th Pontryagin class ''pk''(''E'') is defined as :''pk''(''E'') = ''pk''(''E'', Z) = (−1)''k'' ''c''2''k''(''E'' ⊗ C) ∈ ''H''4''k''(''M'', Z), where: *''c''2''k''(''E'' ⊗ C) denotes the 2''k''-th Chern class of the complexification ''E'' ⊗ C = ''E'' ⊕ ''iE'' of ''E'', *''H''4''k''(''M'', Z) is the 4''k''-cohomology group of ''M'' with integer coefficients. The rational Pontryagin class ''pk''(''E'', Q) is defined to be the image of ''pk''(''E'') in ''H''4''k''(''M'', Q), the 4''k''-cohomology group of ''M'' with rational coefficients. 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「Pontryagin class」の詳細全文を読む スポンサード リンク
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