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

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

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see also **algebraic topology**

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Let

$\langle -,-\rangle
\colon
V \otimes V \to k$

be a bilinear form. A (quadratic) function

$q \colon V \to k$

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

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

for all $v,w \in V$.

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

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

This means that a quadratic refinement by a quadratic form always exists when $2 \in k$ 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:

- Mike Hopkins, Isadore Singer,
*Quadratic Functions in Geometry, Topology, and M-Theory*, J. Diff. Geom.**70**(2005) 329-452 $[$arXiv:math/0211216, euclid.jdg/1143642908$]$

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