L-theory

In mathematics algebraic L-theory is the K-theory of quadratic forms; 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.

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 groups of ε-quadratic forms over the ring R with . More precisely,

is the abelian group of equivalence classes of non-degenerate ε-quadratic forms over R, where the underlying R-modules F are finitely generated free. The equivalence relation is given by stabilization with respect to hyperbolic ε-quadratic forms:

The addition in is defined by

The zero element is represented by for any . The inverse of is .

Odd dimension

Defining odd-dimensional L-groups is more complicated; further details and the definition of the odd-dimensional L-groups can be found in the references mentioned below.