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 formula_1 (Wall) and the symmetric "L"-groups formula_2 (Mishchenko, Ranicki).
Even dimension.
The even-dimensional "L"-groups formula_3 are defined as the Witt groups of ε-quadratic forms over the ring "R" with formula_4. More precisely, 
is the abelian group of equivalence classes formula_6 of non-degenerate ε-quadratic forms formula_7 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 formula_3 is defined by 
The zero element is represented by formula_11 for any formula_12. The inverse of formula_6 is formula_14.
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.
Examples and applications.
The "L"-groups of a group formula_15 are the "L"-groups formula_16 of the group ring formula_17. In the applications to topology formula_15 is the fundamental group
formula_19 of a space formula_20. The quadratic "L"-groups formula_16
play a central role in the surgery classification of the homotopy types of formula_22-dimensional manifolds of dimension formula_23, and in the formulation of the Novikov conjecture.
The distinction between symmetric "L"-groups and quadratic "L"-groups, indicated by upper and lower indices, reflects the usage in group homology and cohomology. The group cohomology formula_24 of the cyclic group formula_25 deals with the fixed points of a formula_25-action, while the group homology formula_27 deals with the orbits of a formula_25-action; compare formula_29 (fixed points) and formula_30 (orbits, quotient) for upper/lower index notation.
The quadratic "L"-groups: formula_31 and the symmetric "L"-groups: formula_32 are related by 
a symmetrization map formula_33 which is an isomorphism modulo 2-torsion, and which corresponds to the polarization identities.
The quadratic and the symmetric "L"-groups are 4-fold periodic (the comment of Ranicki, page 12, on the non-periodicity of the symmetric "L"-groups refers to another type of "L"-groups, defined using "short complexes").
In view of the applications to the classification of manifolds there are extensive calculations of
the quadratic formula_34-groups formula_16. For finite formula_15
algebraic methods are used, and mostly geometric methods (e.g. controlled topology) are used for infinite formula_15. 
More generally, one can define "L"-groups for any additive category with a "chain duality", as in Ranicki (section 1).
Integers.
The simply connected "L"-groups are also the "L"-groups of the integers, as
formula_38 for both formula_34 = formula_40 or formula_41 For quadratic "L"-groups, these are the surgery obstructions to simply connected surgery.
The quadratic "L"-groups of the integers are:
In doubly even dimension (4"k"), the quadratic "L"-groups detect the signature; in singly even dimension (4"k"+2), the "L"-groups detect the Arf invariant (topologically the Kervaire invariant).
The symmetric "L"-groups of the integers are:
In doubly even dimension (4"k"), the symmetric "L"-groups, as with the quadratic "L"-groups, detect the signature; in dimension (4"k"+1), the "L"-groups detect the de Rham invariant.