Contents

# Contents

## Definition

Let

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

be a bilinear form. A 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.

## References

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

Last revised on January 13, 2017 at 14:55:15. See the history of this page for a list of all contributions to it.