L-theory
From Wikipedia, the free encyclopedia
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 very important in surgery theory.
[edit] 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).
The even dimensional L-groups L2k(R) are defined as the Witt groups of ε-quadratic forms over the ring R with ε = ( − 1)k. More precisely, L2k(R) is the abelian group of equivalence classes [ψ] 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: ![[\psi] = [\psi'] \Longleftrightarrow n, n' \in {\mathbb N}_0: \psi \oplus H_{(-1)^k}(R)^n \cong \psi' \oplus H_{(-1)^k}(R)^{n'}](http://upload.wikimedia.org/math/4/d/f/4dfd295adc666308d6ce7a21a9c6035a.png)
.
The addition in L2k(R) is defined by ![[\psi_1] + [\psi_2] := [\psi_1 \oplus \psi_2]](http://upload.wikimedia.org/math/e/6/3/e630d176515ca03653273a6887275dce.png)
. The zero element is represented by 
for any 
. The inverse of [ψ] is [ − ψ].
More details and the definition of the odd dimensional L-groups can be found in the references mentioned below.
[edit] Examples and Applications
The L-groups of a group π are the L-groups ![L_*(\mathbf{Z}[\pi])](http://upload.wikimedia.org/math/2/2/2/2227c4f163a797e4d6ab3a9df59b8b7d.png)
of the group ring ![\mathbf{Z}[\pi]](http://upload.wikimedia.org/math/2/3/4/2347058aaf9de504480efd08b31c1038.png)
. In the applications to topology π is the fundamental group π1(X) of a space X. The quadratic L-groups 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 simply connected L-groups are also the L-groups of the integers: ![L(e) := L(\mathbf{Z}[e]) = L(\mathbf{Z})](http://upload.wikimedia.org/math/f/1/4/f14e4254d1dfbd33c51ab237e469ba0a.png)
with L = L * or L * . For quadratic L-groups, these are the surgery obstructions to simply connected surgery.
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 
deals with the fixed points of a -action, while the group homology H * deals with the orbits of a -action.
The quadratic L-groups: Ln(R) and the symmetric L-groups: Ln(R) 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. The quadratic L-groups of the integers are:
Symmetric L-groups are not 4-periodic in general (see Ranicki, page 12), though they are for the integers. The symmetric L-groups of the integers are:
In view of the applications to the classification of manifolds there are extensive calculations of the quadratic L-groups . 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).
[edit] External links
- Surgery on compact manifolds, by C.T.C. Wall.
- Algebraic L-theory and topological manifolds, by Andrew Ranicki.
- A basic introduction to surgery theory, by Wolfgang Lück.
