# Contents

## Definition

For $V$ a vector space or $k$-module, a quadratic form on $V$ is a function

$q\colon V \to k$

such that for all $v \in V$, $t \in k$

$q(t v) = t^2 q(v)$

and the polarization of $q$

$(v,w) \mapsto q(v+w) - q(v) - q(w)$

is a bilinear form.

Let

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

be a bilinear form. We say a function $q : V \to k$ is a quadratic refinement of $\langle -,-\rangle$ if

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

for all $v,w \in V$.

If this exists, then

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

So a quadratic refinement always exists when $2 \in k$ is invertible. Otherwise its existence is a non-trivial condition.

## References

Course notes include for instance

• On the relation between quadratic and bilinear forms (pdf)

• Bilinear and quadratic forms (pdf)

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

Revised on October 13, 2013 02:56:41 by Urs Schreiber (82.113.121.239)