Signature (topology)

Definition

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 Z. Therefore cup product, under these hypotheses, does give rise to a symmetric bilinear form on H^2k(M,Z); 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.

The signature of M is by definition the signature of Q, an ordered triple according to its definition. If M is not connected, its signature is defined to be the sum of the signatures of its connected components.

Other dimensions

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 L 4 k , or as the 4k-dimensional quadratic L-group L 4 k , and these invariants do not always vanish for other dimensions. The Kervaire invariant is a mod 2 (i.e., an element of Z / 2 ) for framed manifolds of dimension 4k+2 (the quadratic L-group L 4 k + 2 ), while the de Rham invariant is a mod 2 invariant of manifolds of dimension 4k+1 (the symmetric L-group L 4 k + 1 ); the other dimensional L-groups vanish.

Kervaire invariant

When d = 4 k + 2 = 2 ( 2 k + 1 ) is twice an odd integer (singly even), 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. 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. The resulting invariant of a manifold is called the Kervaire invariant.

Properties

René Thom (1954) 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 (1954) found an explicit expression for this linear combination as the L genus of the manifold. William Browder (1962) proved that a simply-connected compact polyhedron with 4n-dimensional Poincaré duality is homotopy equivalent to a manifold if and only if its signature satisfies the expression of the Hirzebruch signature theorem.