In the field of topology, the signature is an integer invariant which is defined for an oriented manifold M of dimension divisible by four.
This invariant of a manifold has been studied in detail, starting with Rokhlin's theorem for 4-manifolds, and Hirzebruch signature theorem.
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 the cup product, under these hypotheses, does give rise to a symmetric bilinear form on H2k(M,R); and therefore to a quadratic form Q.
If M has dimension not divisible by 4, its signature is usually defined to be 0.
There are alternative generalization in L-theory: the signature can be interpreted as the 4k-dimensional (simply connected) symmetric L-group
The Kervaire invariant is a mod 2 (i.e., an element of
) for framed manifolds of dimension 4k+2 (the quadratic L-group
), while the de Rham invariant is a mod 2 invariant of manifolds of dimension 4k+1 (the symmetric L-group
is twice an odd integer (singly even), the same construction gives rise to an antisymmetric bilinear form.
However, if one takes a quadratic refinement of the form, which occurs if one has a framed manifold, then the resulting ε-quadratic forms need not be equivalent, being distinguished by the Arf invariant.