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Concept
L-theory
Summary
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).
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 .
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.
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").
Official source
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