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 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] [2] 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 ${\displaystyle L^{4k},}$ 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

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

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. 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.

See also

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

References

1. Milnor, John; Stasheff, James (1962). Characteristic classes. Annals of Mathematics Studies 246. p. 224. CiteSeerX 10.1.1.448.869. ISBN 978-0691081229.
2. Hatcher, Allen (2003). Algebraic topology (PDF) (Repr. ed.). Cambridge: Cambridge Univ. Pr. p. 250. ISBN 978-0521795401. Retrieved 8 January 2017.
3. Thom, René. "Quelques proprietes globales des varietes differentiables" (PDF) (in French). Comm. Math. Helvetici 28 (1954), S. 17–86. Retrieved 26 October 2019.