nLab
characteristic element of a bilinear form

Context

Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Definition

Definition

For AA an abelian group and given a symmetric bilinear form

,:A×A \langle-,-\rangle \;\colon\; A\times A \longrightarrow \mathbb{Z}

then a characteristic element of the bilinear form is an element λA\lambda\in A such that for all xAx\in A the equation

x,x=x,λmod2 \langle x,x\rangle = \langle x,\lambda \rangle\; mod\;2

holds, or in other words such that

x,x±x,λ \langle x,x\rangle \pm \langle x,\lambda \rangle \in \mathbb{Z}

is divisible by 22\in \mathbb{Z}.

Remark

Often this is considered for the bilinear form being an intersection product in which case AA is a cohomology group and so in this case one also often speaks of characteristic cohomology classes.

Properties

Relation to quadratic refinements

Proposition

A characteristic element determines a quadratic refinement q λq_\lambda of ,\langle-,-\rangle by

q λ:x12(x,x+x,λ). q_\lambda \colon x \mapsto \tfrac{1}{2}\left( \langle x,x\rangle + \langle x,\lambda\rangle \right) \,.
Proof

This follows by trivial direct computation:

q λ(x+y)q λ(x)q λ(y) =12((x 2+y 2+2xy+λx+λy)(x 2+λx)(y 2+λy)) =xy \begin{aligned} q_\lambda(x + y) - q_\lambda(x)- q_\lambda(y) & = \tfrac{1}{2} \left( \left(x^2 + y^2 + 2 x y + \lambda x + \lambda y\right) - \left(x^2 + \lambda x\right) - \left(y^2 + \lambda y\right) \right) \\ & = x y \end{aligned}

the only point being that the prefactor 12\tfrac{1}{2} indeed makes sense, by def. .

Examples

Example

Characteristic elements for the intersection pairing on integral ordinary cohomology are integral lifts of the Wu classes, hence integral Wu structures.

References

Discussion of the refinement of integral Wu structures as the characteristic elements of the intersection product from ordinary cohomology to ordinary differential cohomology is in

Further discussion of the use of characteristic elements in the construction of Theta characteristics is in

  • Bjorn Poonen, Eric Rains, Self cup products and the theta characteristic torsor (arXiv:1104.2105)

Last revised on June 2, 2014 at 10:12:21. See the history of this page for a list of all contributions to it.