Paths and cylinders
For a vector space or more generally a -module, then a quadratic form on is a function
such that for all ,
and the polarization of
is a bilinear form.
be a bilinear form. A function
is called a quadratic refinement of if
for all .
If such is indeed a quadratic form in that then and
This means that a quadratic refinement by a quadratic form always exists when 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.
Course notes include for instance
On the relation between quadratic and bilinear forms (pdf)
Bilinear and quadratic forms (pdf)
section 10 in Analytic theory of modular forms (pdf)
Quadratic refinements of intersection pairing in cohomology is a powerful tool in algebraic topology and differential topology. See
Revised on August 21, 2014 04:48:49
by Urs Schreiber