Editing L-theory (section)

==Examples and applications==
The ''L''-groups of a group <math>\pi</math> are the ''L''-groups <math>L_*(\mathbf{Z}[\pi])</math> of the [[group ring]] <math>\mathbf{Z}[\pi]</math>. In the applications to topology  <math>\pi</math> is the [[fundamental group]]
<math>\pi_1 (X)</math> of a space <math>X</math>. The quadratic ''L''-groups <math>L_*(\mathbf{Z}[\pi])</math>
play a central role in the surgery classification of the homotopy types of <math>n</math>-dimensional [[manifolds]] of dimension <math>n > 4</math>, 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]] <math>H^*</math> of the cyclic group <math>\mathbf{Z}_2</math> deals with the fixed points of a <math>\mathbf{Z}_2</math>-action, while the [[group homology]] <math>H_*</math> deals with the orbits of a <math>\mathbf{Z}_2</math>-action; compare <math>X^G</math> (fixed points) and <math>X_G = X/G</math> (orbits, quotient) for upper/lower index notation.

The quadratic ''L''-groups: <math>L_n(R)</math> and the symmetric ''L''-groups: <math>L^n(R)</math> are related by 
a symmetrization map <math>L_n(R) \to L^n(R)</math> 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 <math>L</math>-groups <math>L_*(\mathbf{Z}[\pi])</math>. For finite <math>\pi</math>
algebraic methods are used, and mostly geometric methods (e.g. controlled topology) are used for infinite <math>\pi</math>. 

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
<math>L(e) := L(\mathbf{Z}[e]) = L(\mathbf{Z})</math> for both <math>L</math> = <math>L^*</math> or <math>L_*.</math> For quadratic ''L''-groups, these are the surgery obstructions to [[simply connected]] surgery.

The quadratic ''L''-groups of the integers are:
:<math>\begin{align}
L_{4k}(\mathbf{Z}) &= \mathbf{Z}   && \text{signature}/8\\
L_{4k+1}(\mathbf{Z}) &= 0\\
L_{4k+2}(\mathbf{Z}) &= \mathbf{Z}/2 && \text{Arf invariant}\\
L_{4k+3}(\mathbf{Z}) &= 0.
\end{align}</math>
In [[doubly even]] dimension (4''k''), the quadratic ''L''-groups detect the [[signature (topology)|signature]]; in [[singly even]] dimension (4''k''+2), the ''L''-groups detect the [[Arf invariant]] (topologically the [[Kervaire invariant]]).

The symmetric ''L''-groups of the integers are:
:<math>\begin{align}
L^{4k}(\mathbf{Z}) &= \mathbf{Z} && \text{signature}\\
L^{4k+1}(\mathbf{Z}) &= \mathbf{Z}/2 && \text{de Rham invariant}\\
L^{4k+2}(\mathbf{Z}) &= 0\\
L^{4k+3}(\mathbf{Z}) &= 0.
\end{align}</math>
In doubly even dimension (4''k''), the symmetric ''L''-groups, as with the quadratic ''L''-groups, detect the signature; in dimension (4''k''+1), the ''L''-groups detect the [[de Rham invariant]].