| Revision as of 10:49, 17 December 2008 editRanicki (talk | contribs)Extended confirmed users602 edits →See also← Previous edit | Revision as of 08:16, 26 December 2008 edit undoRanicki (talk | contribs)Extended confirmed users602 editsNo edit summaryNext edit → | ||
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| :<math>\alpha^p \smile \beta^q = (-1)^{pq}(\beta^q \smile \alpha^p)</math> | :<math>\alpha^p \smile \beta^q = (-1)^{pq}(\beta^q \smile \alpha^p)</math> | ||
| shows that with ''p'' = ''q'' = 2''n'' the product is ]. It takes values in | shows that with ''p'' = ''q'' = 2''n'' the product is ]. It takes values in | ||
| :''H''<sup>4''n''</sup>(''M'',''R''). | :''H''<sup>4''n''</sup>(''M'',''R''). | ||
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| The ] 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 ]. This invariant of a manifold has been studied in detail, starting with ] for 4-manifolds. | The ] 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 ]. This invariant of a manifold has been studied in detail, starting with ] for 4-manifolds. | ||
| When ''d'' is ], 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 ], 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. | ||
| ] (1954) showed that the signature of a manifold is a cobordism invariant, and in particular is given by some linear combination of its ] numbers. ] (1954) found an explicit expression for this linear combination as the ] of the manifold. (1962) proved that a simply-connected compact ] with ''4n''-dimensional ] is homotopy equivalent to a manifold if and only if its signature |
] (1954) showed that the signature of a manifold is a cobordism invariant, and in particular is given by some linear combination of its ] numbers. ] (1954) found an explicit expression for this linear combination as the ] of the manifold. (1962) proved that a simply-connected compact ] with ''4n''-dimensional ] is homotopy equivalent to a manifold if and only if its signature satisfies the expression of the ] | ||
| ==See also== | ==See also== | ||
Revision as of 08:16, 26 December 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 symmetric. 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. 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. 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 (1954) 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 (1954) found an explicit expression for this linear combination as the L genus of the manifold. William Browder (1962) proved that a simply-connected compact polyhedron with 4n-dimensional Poincaré duality is homotopy equivalent to a manifold if and only if its signature satisfies the expression of the Hirzebruch signature theorem