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.
One can define L-groups for any ring with involution R: the quadratic L-groups (Wall) and the symmetric L-groups (Mishchenko, Ranicki).
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 .
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).
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:
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.
- Lück, Wolfgang (2002), "A basic introduction to surgery theory", Topology of high-dimensional manifolds, No. 1, 2 (Trieste, 2001) (PDF), ICTP Lect. Notes, 9, Abdus Salam Int. Cent. Theoret. Phys., Trieste, pp. 1–224, MR 1937016
- Ranicki, Andrew A. (1992), Algebraic L-theory and topological manifolds (PDF), Cambridge Tracts in Mathematics, 102, Cambridge University Press, ISBN 978-0-521-42024-2, MR 1211640
- Wall, C. T. C. (1999) , Ranicki, Andrew (ed.), Surgery on compact manifolds (PDF), Mathematical Surveys and Monographs, 69 (2nd ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0942-6, MR 1687388