Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
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
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related lectures (1)
Acyclic Models: Cup Product and Cohomology
Covers the cup product on cohomology, acyclic models, and the universal coefficient theorem.
Related publications (6)
Invariant forms on irreducible modules of simple algebraic groups
Let G be a simple linear algebraic group over an algebraically dosed field K of characteristic p >= 0 and let V be an irreducible rational G-module with highest weight A. When is self-dual, a basic question to ask is whether V has a non-degenerate G-invari ...
Elsevier2017Related concepts (6)
Ε-quadratic form
In mathematics, specifically the theory of quadratic forms, an ε-quadratic form is a generalization of quadratic forms to skew-symmetric settings and to *-rings; ε = ±1, accordingly for symmetric or skew-symmetric. They are also called -quadratic forms, particularly in the context of surgery theory. There is the related notion of ε-symmetric forms, which generalizes symmetric forms, skew-symmetric forms (= symplectic forms), Hermitian forms, and skew-Hermitian forms.
Arf invariant
In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by Turkish mathematician when he started the systematic study of quadratic forms over arbitrary fields of characteristic 2. The Arf invariant is the substitute, in characteristic 2, for the discriminant for quadratic forms in characteristic not 2. Arf used his invariant, among others, in his endeavor to classify quadratic forms in characteristic 2.
Genus of a multiplicative sequence
In mathematics, a genus of a multiplicative sequence is a ring homomorphism from the ring of smooth compact manifolds up to the equivalence of bounding a smooth manifold with boundary (i.e., up to suitable cobordism) to another ring, usually the rational numbers, having the property that they are constructed from a sequence of polynomials in characteristic classes that arise as coefficients in formal power series with good multiplicative properties.