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Revision as of 11:07, 12 August 2009 editTea2min (talk | contribs)Extended confirmed users, Pending changes reviewers21,806 editsm compact→compact manifold← Previous edit Revision as of 16:20, 22 October 2009 edit undo18.111.60.181 (talk)No edit summaryNext edit →
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be defined in this way for any general compact ] with ''4n''-dimensional ]. be defined in this way for any general compact ] with ''4n''-dimensional ].


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 '''signature''' of ''M'' is by definition the ] 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.

Revision as of 16:20, 22 October 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

α p ⌣ β q = ( − 1 ) p q ( β q ⌣ α p ) {\displaystyle \alpha ^{p}\smile \beta ^{q}=(-1)^{pq}(\beta ^{q}\smile \alpha ^{p})}

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

See also

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