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 Edit

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


The basic identity for the cup product


shows that with p = q = 2k 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 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   of M is by definition the signature of Q, that is,   where any diagonal matrix defining Q has   positive entries and   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 Edit

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

Kervaire invariant Edit

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.

Properties Edit

  • Compact oriented manifolds M and N satisfy   by definition, and satisfy   by a Künneth formula.
  • If M is an oriented boundary, then  .
  • 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.

See also Edit

References Edit

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