In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, algebraic ''L''-theory is the
''K''-theory of
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
s; the term was coined by
C. T. C. Wall,
with ''L'' being used as the letter after ''K''. Algebraic ''L''-theory, also known as "Hermitian ''K''-theory",
is important in
surgery theory
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by . Milnor called this technique ''surgery'', while And ...
.
Definition
One can define ''L''-groups for any
ring with involution ''R'': the quadratic ''L''-groups
(Wall) and the symmetric ''L''-groups
(Mishchenko, Ranicki).
Even dimension
The even-dimensional ''L''-groups
are defined as the
Witt group
In mathematics, a Witt group of a field, named after Ernst Witt, is an abelian group whose elements are represented by symmetric bilinear forms over the field.
Definition
Fix a field ''k'' of characteristic not equal to two. All vector spaces ...
s of
ε-quadratic forms over the ring ''R'' with
. More precisely,
::
is the abelian group of equivalence classes