Editing L-theory (section)

==Definition==
One can define ''L''-groups for any [[ring with involution]] ''R'': the quadratic ''L''-groups <math>L_*(R)</math> (Wall) and the symmetric ''L''-groups <math>L^*(R)</math> (Mishchenko, Ranicki).

=== Even dimension ===
The even-dimensional ''L''-groups <math>L_{2k}(R)</math> are defined as the [[Witt group]]s of [[ε-quadratic forms]] over the ring ''R'' with <math>\epsilon = (-1)^k</math>. More precisely, 

::<math>L_{2k}(R)</math> 

is the abelian group of equivalence classes <math>[\psi]</math> of non-degenerate ε-quadratic forms <math>\psi \in Q_\epsilon(F)</math> 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]]: 

:<math>[\psi] = [\psi'] \Longleftrightarrow n, n' \in {\mathbb N}_0: \psi \oplus H_{(-1)^k}(R)^n \cong \psi' \oplus H_{(-1)^k}(R)^{n'}</math>.

The addition in <math>L_{2k}(R)</math> is defined by 

:<math>[\psi_1] + [\psi_2] := [\psi_1 \oplus \psi_2].</math>

The zero element is represented by <math>H_{(-1)^k}(R)^n</math> for any <math>n \in {\mathbb N}_0</math>. The inverse of <math>[\psi]</math> is <math>[-\psi]</math>.

=== 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.