Signature (topology)
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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
- H2n(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
- H4n(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 H2n(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.
Thom showed that the signature of a manifold is a cobordism invariant, and in particular is given by some linear combination of it Pontryagin numbers. Hirzebruch found an explicit expression for this linear combination as the L genus of the manifold.
