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
Quadratic refinements of intersection pairing in cohomology is a powerful tool in algebraic topology and differential topology. See
Revised on June 2, 2014 22:26:46
by Urs Schreiber