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Concept
Signature (topology)
Summary
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.
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. 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. If M is not connected, its signature is defined to be the sum of the signatures of its connected components.
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
When is twice an odd integer (singly even), the same construction gives rise to an antisymmetric bilinear form.
Official source
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