| Revision as of 05:40, 15 December 2006 editTurgidson (talk | contribs)Extended confirmed users61,660 edits links to Thom, Pontryagin, Hirzebruch← Previous edit | Revision as of 11:34, 24 February 2008 edit undoMelchoir (talk | contribs)Extended confirmed users32,110 editsm divisible by fourNext edit → | ||
| Line 1: | Line 1: | ||
| In ], the '''signature''' of an oriented ] ''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 an oriented ] ''M'' is defined when ''M'' has dimension ''d'' ]. 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''), | ||
| Line 25: | Line 25: | ||
| When ''d'' is twice an odd integer, the same construction gives rise to an ]. Such forms do not have a signature invariant; if they are non-degenerate, any two such forms are equivalent. | When ''d'' is twice an odd integer, the same construction gives rise to an ]. Such forms do not have a signature invariant; if they are non-degenerate, any two such forms are equivalent. | ||
| ] showed that the signature of a manifold is a cobordism invariant, and in particular is given by some linear combination of its ] numbers. ] found an explicit expression for this linear combination as the ] of the manifold. |
] showed that the signature of a manifold is a cobordism invariant, and in particular is given by some linear combination of its ] numbers. ] found an explicit expression for this linear combination as the ] of the manifold. | ||
| ==See also== | ==See also== | ||
Revision as of 11:34, 24 February 2008
In mathematics, the signature of an oriented 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 M is by definition the signature of Q. If M is not connected, its signature is defined to be the sum of the signatures of its connected components. If M has dimension not divisible by 4, its signature is usually defined to be 0. The form Q is non-degenerate. This invariant of a manifold has been studied in detail, starting with Rokhlin's theorem for 4-manifolds.
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.
René Thom showed that the signature of a manifold is a cobordism invariant, and in particular is given by some linear combination of its Pontryagin numbers. Friedrich Hirzebruch found an explicit expression for this linear combination as the L genus of the manifold.