Editing L-theory (section)

=== 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]].