| Revision as of 12:03, 10 November 2004 editCharles Matthews (talk | contribs)Autopatrolled, Administrators360,392 edits initial page | Revision as of 20:40, 23 May 2005 edit undoNowhither (talk | contribs)1,328 editsm Fixed "connected" linkNext edit → | ||
| Line 1: | Line 1: | ||
| In ], the '''signature''' of a ] ''M'' is defined when ''M'' has dimension ''d'' divisible by four. In that case, when ''M'' is ] and ], ] gives rise to a ] ''Q'' on the 'middle' real ] | In ], the '''signature''' of a ] ''M'' is defined when ''M'' has dimension ''d'' divisible by four. In that case, when ''M'' is ] and ], ] gives rise to a ] ''Q'' on the 'middle' real ] | ||
| :''H''<sup>2''n''</sup>(''M'',''R''), | :''H''<sup>2''n''</sup>(''M'',''R''), | ||
Revision as of 20:40, 23 May 2005
In mathematics, the signature of a manifold M is defined when M has dimension d divisible by four. In that case, when M is connected and orientable, cup product gives rise to a quadratic form Q on the 'middle' real cohomology group
- H(M,R),
where
- d = 4n.
The basic identity for the cup product
shows that with p = q = 2n the product is commutative. It takes values in
- H(M,R).
If we assume also that M is compact, Poincaré duality identifies this with
- H0(M,R),
which is a one-dimensional real vector space and can be identified with R. Therefore cup product, under these hypotheses, does give rise to a symmetric bilinear form on H(M,R); and therefore to a quadratic form Q.
The signature of Q is by definition the signature of M. It can be shown that Q is non-degenerate. This invariant of a manifold has been studied in detail, starting with work of Rokhlin. Further invariants of Q as an integral quadratic form are also of interest in topology.
When d is twice an odd integer, the same construction gives rise to an antisymmetric bilinear form. Such forms do not have a signature invariant; if they are non-degenerate, any two such forms are equivalent.
Categories: