nLab quadratic refinement

Definition

Let

$$\langle -,-\rangle \colon V \otimes V \to k$$

be a bilinear form. A (quadratic) function

$$q \colon V \to k$$

is called a quadratic refinement of $\langle -,-\rangle$ if

$$\langle v,w\rangle = q(v + w) - q(v) - q(w) + q(0)$$

for all $v,w \in V$.

If such $q$ is indeed a quadratic form in that $q(t v) = t^2 q(v)$ then $q(0) = 0$ and

$$\langle v , v \rangle = 2 q(v) \,.$$

This means that a quadratic refinement by a quadratic form always exists when $2 \in k$ is invertible. Otherwise its existence is a non-trivial condition. One way to express quadratic refinements is by characteristic elements of a bilinear form. See there for more.

Related concepts

• cubical structure on a line bundle

References

Quadratic refinements of intersection pairing in cohomology is a powerful tool in algebraic topology and differential topology. See:

• Mike Hopkins, Isadore Singer, Quadratic Functions in Geometry, Topology, and M-Theory, J. Diff. Geom. 70 (2005) 329-452 $[$arXiv:math/0211216, euclid.jdg/1143642908$]$