Signature (topology)
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.
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 groupThe 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; 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, that is, where any diagonal matrix defining Q has positive entries and negative entries. 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 symmetric L-group or as the 4k-dimensional quadratic L-group and these invariants do not always vanish for other dimensions. The Kervaire invariant is a mod 2 for framed manifolds of dimension 4k+2, while the de Rham invariant is a mod 2 invariant of manifolds of dimension 4k+1 ; the other dimensional L-groups vanish.Kervaire invariant
When 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. 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
- Compact oriented manifolds M and N satisfy by definition, and satisfy by a Künneth formula.
- If M is an oriented boundary, then.
- René Thom showed that the signature of a manifold is a cobordism invariant, and in particular is given by some linear combination of its Pontryagin numbers. For example, in four dimensions, it is given by. Friedrich Hirzebruch found an explicit expression for this linear combination as the L genus of the manifold.
- William Browder 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.
- Rokhlin's theorem says that the signature of a 4-dimensional simply connected manifold with a spin structure is divisible by 16.