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In the mathematical field of topology, the signature is an integer invariant which is defined for an oriented manifold ''M'' of dimension ''d''=4''k'' divisible by four (doubly even-dimensional). This invariant of a manifold has been studied in detail, starting with Rokhlin's theorem for 4-manifolds. == Definition == Given a connected and oriented manifold ''M'' of dimension 4''k'', the cup product gives rise to a quadratic form ''Q'' on the 'middle' real cohomology group :''H''2''k''(''M'',''Z''). The basic identity for the cup product : shows that with ''p'' = ''q'' = 2''k'' the product is symmetric. It takes values in :''H''4''k''(''M'',''Z''). If we assume also that ''M'' is compact, Poincaré duality identifies this with :''H''0(''M'',''Z''), which can be identified with ''Z''. Therefore cup product, under these hypotheses, does give rise to a symmetric bilinear form on ''H''2''k''(''M'',''Z''); and therefore to a quadratic form ''Q''. The form ''Q'' is non-degenerate due to Poincaré duality, as it pairs non-degenerately with itself. More generally, the signature can be defined in this way for any general compact polyhedron with ''4n''-dimensional Poincaré duality. The signature of ''M'' is by definition the signature of ''Q'', an ordered triple according to its definition. If ''M'' is not connected, its signature is defined to be the sum of the signatures of its connected components. 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「Signature (topology)」の詳細全文を読む スポンサード リンク
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