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.

## Contents

* 1 Definition
* 2 Other dimensions
* 2.1 Kervaire invariant
* 3 Properties
* 4 See also
* 5 References

## 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

H2k(M,𝐑).

The basic identity for the cup product

αp⌣βq=(−1)pq(βq⌣αp)

shows that with p = q = 2k the product is symmetric. It takes values in

H4k(M,𝐑).

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

H0(M,𝐑)

which can be identified with 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 σ(M) of M is by definition the signature of Q, that is, σ(M)=n+−n− where any diagonal matrix defining Q has n+ positive entries and 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

Further information: L-theory

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 L4k, or as the 4k-dimensional quadratic L-group L4k, 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 L4k+2), while the de Rham invariant is a mod 2 invariant of manifolds of dimension 4k+1 (the symmetric L-group L4k+1); the other dimensional L-groups vanish.

### Kervaire invariant

Main article: Kervaire invariant

When 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 σ(M⊔N)=σ(M)+σ(N) by definition, and satisfy σ(M×N)=σ(M)σ(N) by a Künneth formula.

* If M is an oriented boundary, then σ(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 p13. 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.

* Rokhlin's theorem says that the signature of a 4-dimensional simply connected manifold with a spin structure is divisible by 16.

## See also

* Hirzebruch signature theorem
* Genus of a multiplicative sequence
* Rokhlin's theorem

## References

1. ↑ Hatcher, Allen (2003) (in en). Algebraic topology (Repr. ed.). Cambridge: Cambridge Univ. Pr.. p. 250. ISBN 978-0521795401. https://www.math.cornell.edu/~hatcher/AT/AT.pdf. Retrieved 8 January 2017.
2. ↑ Milnor, John; Stasheff, James (1962) (in en). Characteristic classes. Annals of Mathematics Studies 246. p. 224. ISBN 978-0691081229.
3. ↑ Thom, René. "Quelques proprietes globales des varietes differentiables" (in fr). Comm. Math. Helvetici 28 (1954), S. 17–86. https://www.maths.ed.ac.uk/~v1ranick/papers/thomcob.pdf. Retrieved 26 October 2019.

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