# Contents

## Definition

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

$q:V\to k$q\colon V \to k

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

$q\left(tv\right)={t}^{2}q\left(v\right)$q(t v) = t^2 q(v)

and the polarization of $q$

$\left(v,w\right)↦q\left(v+w\right)-q\left(v\right)-q\left(w\right)$(v,w) \mapsto q(v+w) - q(v) - q(w)

is a bilinear form.

Let

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

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

$⟨v,w⟩=q\left(v+w\right)-q\left(v\right)-q\left(w\right)$\langle v,w\rangle = q(v + w) - q(v) - q(w)

for all $v,w\in V$.

If this exists, then

$⟨v,v⟩=2q\left(v\right)\phantom{\rule{thinmathspace}{0ex}}.$\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)