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 and the symmetric L-groups .

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.

Examples and applications

The L-groups of a group are the L-groups of the group ring. In the applications to topology is the fundamental group
of a space. The quadratic L-groups
play a central role in the surgery classification of the homotopy types of -dimensional manifolds of dimension, 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 of the cyclic group deals with the fixed points of a -action, while the group homology deals with the orbits of a -action; compare and for upper/lower index notation.
The quadratic L-groups: and the symmetric L-groups: are related by
a symmetrization map 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.
In view of the applications to the classification of manifolds there are extensive calculations of
the quadratic -groups. For finite
algebraic methods are used, and mostly geometric methods are used for infinite.
More generally, one can define L-groups for any additive category with a chain duality, as in Ranicki.

Integers

The simply connected L-groups are also the L-groups of the integers, as
for both = or 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, the quadratic L-groups detect the signature; in singly even dimension, the L-groups detect the Arf invariant.
The symmetric L-groups of the integers are:
In doubly even dimension, the symmetric L-groups, as with the quadratic L-groups, detect the signature; in dimension, the L-groups detect the de Rham invariant.