L-theory

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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[edit]

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[edit]

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[edit]

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[edit]

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 L-groups are 4-fold periodic. Symmetric L-groups are not 4-periodic in general (see Ranicki, page 12), though they are for the integers.

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[edit]

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 (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.

References[edit]