L-theory

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Algebraic L-theory is the K-theory of quadratic forms; the term was coined by C. T. C. Wall, and the theory is very important in surgery theory.

[edit] Definition

One can define L-groups for any ring with involution.

The L-groups of a group $π$ are the L-groups of the group ring $\mathbf{Z}[\pi]$ . $π$ is used to denote the group because in surgery theory one uses the group $π 1 X$ .

The simply connected L-groups are also the L-groups of the integers: $L(e) := L(\mathbf{Z}[e]) = L(\mathbf{Z})$ (this notation holds for both quadratic and symmetric L-groups). For quadratic L-groups, these are the surgery obstructions to simply connected surgery.

More generally, one can define L-groups for any additive category with a chain duality, as in Ranicki (section 1).

There is a distinction between symmetric L-groups and quadratic L-groups, indicated by upper and lower indices.

Quadratic L-groups: $L n (R)$ .

Symmetric L-groups: $L n (R)$ .

These are related by a symmetrization map $L_n(R) \to L^n(R)$ , which corresponds to the polarization identities.

The quadratic L-groups are 4-fold periodic. The quadratic L-groups of the integers are:

$\begin{align} L_{4k}(\mathbf{Z}) &= \mathbf{Z} && \mbox{signature}\\ L_{4k+1}(\mathbf{Z}) &= 0\\ L_{4k+2}(\mathbf{Z}) &= \mathbf{Z}/2 && \mbox{Arf invariant}\\ L_{4k+3}(\mathbf{Z}) &= 0 \end{align}$

Symmetric L-groups are not 4-periodic in general (p. 12), though they are for the integers. The symmetric L-groups of the integers are:

$\begin{align} L_{4k}(\mathbf{Z}) &= \mathbf{Z}\\ L_{4k+1}(\mathbf{Z}) &= \mathbf{Z}/2\\ L_{4k+2}(\mathbf{Z}) &= 0\\ L_{4k+3}(\mathbf{Z}) &= 0 \end{align}$