# 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.

## DefinitionEdit

Given a connected and oriented manifold *M* of dimension 4*k*, the cup product gives rise to a quadratic form *Q* on the 'middle' real cohomology group

- .

The basic identity for the cup product

shows that with *p* = *q* = 2*k* 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 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 dimensionsEdit

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 4*k*-dimensional (simply-connected) symmetric L-group or as the 4*k*-dimensional quadratic L-group and these invariants do not always vanish for other dimensions. The Kervaire invariant is a mod 2 (i.e., an element of ) for framed manifolds of dimension 4*k*+2 (the quadratic L-group ), while the de Rham invariant is a mod 2 invariant of manifolds of dimension 4*k*+1 (the symmetric L-group ); the other dimensional L-groups vanish.

### Kervaire invariantEdit

When 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.

## PropertiesEdit

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 . 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 4*n*-dimensional Poincaré duality is homotopy equivalent to a manifold if and only if its signature satisfies the expression of the Hirzebruch signature theorem.

## See alsoEdit

## ReferencesEdit

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