Paths and cylinders
For a vector space or -module, a quadratic form on is a function
q\colon V \to k
such that for all ,
q(t v) = t^2 q(v)
and the polarization of
(v,w) \mapsto q(v+w) - q(v) - q(w)
is a bilinear form.
V \otimes V \to k
be a bilinear form. We say a function is a quadratic refinement of if
\langle v,w\rangle =
q(v + w) - q(v) - q(w)
for all .
If this exists, then
\langle v , v \rangle = 2 q(v)
So a quadratic refinement always exists when is invertible. Otherwise its existence is a non-trivial condition.
Course notes include for instance
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