nLab characteristic element of a bilinear form

Definition

Definition

For $A$ an abelian group and given a symmetric bilinear form

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

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

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

holds, or in other words such that

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

is divisible by $2\in \mathbb{Z}$.

Remark

Often this is considered for the bilinear form being an intersection product in which case $A$ 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 $q_\lambda$ of $\langle-,-\rangle$ by

$$q_\lambda \colon x \mapsto \tfrac{1}{2}\left( \langle x,x\rangle + \langle x,\lambda\rangle \right) \,.$$

Proof

This follows by trivial direct computation:

$$\begin{aligned} q_\lambda(x + y) - q_\lambda(x)- q_\lambda(y) & = \tfrac{1}{2} \left( \left(x^2 + y^2 + 2 x y + \lambda x + \lambda y\right) - \left(x^2 + \lambda x\right) - \left(y^2 + \lambda y\right) \right) \\ & = x y \end{aligned}$$

the only point being that the prefactor $\tfrac{1}{2}$ 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

• Michael Hopkins, Isadore Singer, Quadratic Functions in Geometry, Topology, and M-Theory, (arXiv:math.AT/0211216)

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)