Signature of a manifold
In the mathematical field of topology, the signature is an integer invariant which is defined for an oriented manifold M of dimension d=4k divisible by four (doubly even-dimensional).
This invariant of a manifold has been studied in detail, starting with Rokhlin's theorem for 4-manifolds.
Given a connected and oriented manifold M of dimension 4k, the cup product gives rise to a quadratic form Q on the 'middle' real cohomology group
The basic identity for the cup product
shows that with p = q = 2k the product is symmetric. It takes values in
If we assume also that M is compact, Poincaré duality identifies this with
which can be identified with . Therefore cup product, under these hypotheses, does give rise to a symmetric bilinear form on H2k(M,R); and therefore to a quadratic form Q. The form Q is non-degenerate due to Poincaré duality, as it pairs non-degenerately with itself. More generally, the signature can be defined in this way for any general compact polyhedron with 4n-dimensional Poincaré duality.
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