L theory

Definition

One can define L-groups for any ring with involution R: the quadratic L-groups L ∗ ( R ) (Wall) and the symmetric L-groups L ∗ ( R ) (Mishchenko, Ranicki).

Even dimension

The even-dimensional L-groups L 2 k ( R ) are defined as the Witt groups of ε-quadratic forms over the ring R with ϵ = ( − 1 ) k . More precisely,

is the abelian group of equivalence classes [ ψ ] of non-degenerate ε-quadratic forms ψ ∈ Q ϵ ( F ) 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 L 2 k ( R ) is defined by

The zero element is represented by H ( − 1 ) k ( R ) n for any n ∈ N 0 . The inverse of [ ψ ] is [ − ψ ] .

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 π are the L-groups L ∗ ( Z [ π ] ) of the group ring Z [ π ] . In the applications to topology π is the fundamental group π 1 ( X ) of a space X . The quadratic L-groups L ∗ ( Z [ π ] ) play a central role in the surgery classification of the homotopy types of n -dimensional manifolds of dimension n > 4 , 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 H ∗ of the cyclic group Z 2 deals with the fixed points of a Z 2 -action, while the group homology H ∗ deals with the orbits of a Z 2 -action; compare X G (fixed points) and X G = X / G (orbits, quotient) for upper/lower index notation.

The quadratic L-groups: L n ( R ) and the symmetric L-groups: L n ( R ) are related by a symmetrization map L n ( R ) → L n ( R ) 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 L -groups L ∗ ( Z [ π ] ) . For finite π algebraic methods are used, and mostly geometric methods (e.g. controlled topology) are used for infinite π .

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 L ( e ) := L ( Z [ e ] ) = L ( Z ) for both L = L ∗ or L ∗ . 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.