Signature (topology)

In mathematics, the signature of a 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

${\displaystyle \alpha ^{p}\smile \beta ^{q}=(-1)^{pq}(\beta ^{q}\smile \alpha ^{p})}$

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 Q is by definition the signature of M. It can be shown that Q is non-degenerate. This invariant of a manifold has been studied in detail, starting with work of Rokhlin. Further invariants of Q as an integral quadratic form are also of interest in topology.

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.