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] One can define L-groups for any ring with involution R: the quadratic L-groups
(Wall) and the symmetric 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
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.
is the fundamental group
play a central role in the surgery classification of the homotopy types of
, 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
(fixed points) and
(orbits, quotient) for upper/lower index notation.
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
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).
The simply connected L-groups are also the L-groups of the integers, as
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 (4k), the quadratic L-groups detect the signature; in singly even dimension (4k+2), the L-groups detect the Arf invariant (topologically the Kervaire invariant).
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.