nLab quadratic refinement



Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts





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

be a bilinear form. A (quadratic) function

q:Vk q \colon V \to k

is called a quadratic refinement of ,\langle -,-\rangle if

v,w=q(v+w)q(v)q(w)+q(0) \langle v,w\rangle = q(v + w) - q(v) - q(w) + q(0)

for all v,wVv,w \in V.

If such qq is indeed a quadratic form in that q(tv)=t 2q(v)q(t v) = t^2 q(v) then q(0)=0q(0) = 0 and

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

This means that a quadratic refinement by a quadratic form always exists when 2k2 \in k 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.


Quadratic refinements of intersection pairing in cohomology is a powerful tool in algebraic topology and differential topology. See:

Last revised on May 7, 2022 at 19:59:52. See the history of this page for a list of all contributions to it.