quadratic form



For VV a vector space or kk-module, a quadratic form on VV is a function

q:Vk q\colon V \to k

such that for all vVv \in V, tkt \in k

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

and the polarization of qq

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

is a bilinear form.

Quadratic refinement


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

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

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

for all v,wVv,w \in V.

If this exists, then

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

So a quadratic refinement always exists when 2k2 \in k is invertible. Otherwise its existence is a non-trivial condition.


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 (