# 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 group

$\displaystyle{ H^{2k}(M,\mathbf{R}) }$.

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 = 2k the product is symmetric. It takes values in

$\displaystyle{ H^{4k}(M,\mathbf{R}) }$.

If we assume also that M is compact, Poincaré duality identifies this with

$\displaystyle{ H^{0}(M,\mathbf{R}) }$

which can be identified with $\displaystyle{ \mathbf{R} }$. 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. The form Q is non-degenerate due to Poincaré duality, as it pairs non-degenerately with itself.[1] More generally, the signature can be defined in this way for any general compact polyhedron with 4n-dimensional Poincaré duality.

The signature $\displaystyle{ \sigma(M) }$ of M is by definition the signature of Q, that is, $\displaystyle{ \sigma(M) = n_+ - n_- }$ where any diagonal matrix defining Q has $\displaystyle{ n_+ }$ positive entries and $\displaystyle{ n_- }$ negative entries.[2] 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 $\displaystyle{ L^{4k}, }$ or as the 4k-dimensional quadratic L-group $\displaystyle{ L_{4k}, }$ and these invariants do not always vanish for other dimensions. The Kervaire invariant is a mod 2 (i.e., an element of $\displaystyle{ \mathbf{Z}/2 }$) for framed manifolds of dimension 4k+2 (the quadratic L-group $\displaystyle{ L_{4k+2} }$), while the de Rham invariant is a mod 2 invariant of manifolds of dimension 4k+1 (the symmetric L-group $\displaystyle{ L^{4k+1} }$); the other dimensional L-groups vanish.

### Kervaire invariant

Main page: Kervaire invariant

When $\displaystyle{ d=4k+2=2(2k+1) }$ 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

• Compact oriented manifolds M and N satisfy $\displaystyle{ \sigma(M \sqcup N) = \sigma(M) + \sigma(N) }$ by definition, and satisfy $\displaystyle{ \sigma(M\times N) = \sigma(M)\sigma(N) }$ by a Künneth formula.
• If M is an oriented boundary, then $\displaystyle{ \sigma(M)=0 }$.
• 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.[3] For example, in four dimensions, it is given by $\displaystyle{ \frac{p_1}{3} }$. Friedrich Hirzebruch (1954) found an explicit expression for this linear combination as the L genus of the manifold.