Signature (topology) explained

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

H^2k(M,R)

The basic identity for the cup product

\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

H^4k(M,R)

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

H⁰(M,R)

which can be identified with

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

\sigma(M)

of M is by definition the signature of Q, that is,

\sigma(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

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^4k,

or as the 4k-dimensional quadratic L-group

L_4k,

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_4k+2

), while the de Rham invariant is a mod 2 invariant of manifolds of dimension 4k+1 (the symmetric L-group

L^4k+1

); the other dimensional L-groups vanish.

Kervaire invariant

See 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

\sigma(M\sqcupN)=\sigma(M)+\sigma(N)

by definition, and satisfy

\sigma(M x N)=\sigma(M)\sigma(N)

by a Künneth formula.

If M is an oriented boundary, then

\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

	p₁
	3

. 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.

Notes and References

Book: Hatcher. Allen. Algebraic topology. 2003. Cambridge Univ. Pr.. Cambridge. 978-0521795401. 250. Repr.. 8 January 2017. en.
Book: Milnor. John. Stasheff. James. Characteristic classes. 1962. Annals of Mathematics Studies 246. 224. 978-0691081229. en. 10.1.1.448.869.
News: Thom. René. Quelques proprietes globales des varietes differentiables. Comm. Math. Helvetici 28 (1954), S. 17–86. 26 October 2019. fr.