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In mathematics, the Atiyah–Hirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by in the special case of topological K-theory. For a CW complex ''X'', it relates the generalized cohomology groups : ''h''''i''(''X'') with 'ordinary' cohomology groups ''H'' ''j'' with coefficients in the generalized cohomology of a point. More precisely, the E2 term of the spectral sequence is H''i''(''X'',''h''''j''(point)), and the spectral sequence converges conditionally to ''h''''i''+''j''(''X''). Atiyah and Hirzebruch pointed out a generalization of their spectral sequence that also generalizes the Serre spectral sequence, and reduces to it in the case where ''h''=''H''. It can be derived from an exact couple that gives the ''E''1 page of the Serre spectral sequence, except with the ordinary cohomology groups replaced with ''h''. In detail, assume ''X'' to be the total space of a Serre fibration with fibre ''F'' and base space ''B''. The filtration of ''B'' by its ''n''-skeletons gives rise to a filtration of ''X''. There is a corresponding spectral sequence with ''E''2 term : ''H''''p''(''B'';''h'' ''q''(''F'')) and abutting to the associated graded ring of the filtered ring : ''h''''p'' + ''q''(''X''). This is the Atiyah–Hirzebruch spectral sequence in the case where the fibre ''F'' is a point. ==References== * 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「Atiyah–Hirzebruch spectral sequence」の詳細全文を読む スポンサード リンク
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