Signature (topology)

Signature (topology)

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"2"n"("M","R"),

where

:"d" = 4"n".

The basic identity for the cup product

:alpha^p smile eta^q = (-1)^{pq}(eta^q smile alpha^p)

shows that with "p" = "q" = 2"n" the product is commutative. It takes values in

:"H"4"n"("M","R").

If we assume also that "M" is compact, Poincaré duality identifies this with

:"H"0("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"2"n"("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.

ee also

*Hirzebruch signature theorem
*Genus of a multiplicative sequence


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