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| 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 ] | 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''), | ||
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| 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 satisfies the expression of the ] | ] (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 22:52, 6 April 2009
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