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.[1]
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
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
(fixed points) and
(orbits, quotient) 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 (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 -groups . For finite
algebraic methods are used, and mostly geometric methods (e.g. controlled topology) are used for infinite .
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
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
).
The symmetric L-groups of the integers are:
In doubly even dimension (4k), the symmetric L-groups, as with the quadratic L-groups, detect the signature; in dimension (4k+1), the L-groups detect the de Rham invariant.
References