L-theory

In mathematics, 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 important in surgery 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 $$L_{2k}(R)$$ are defined as the Witt groups of ε-quadratic forms over the ring R with $$\epsilon = (-1)^k$$. More precisely,


 * $$L_{2k}(R)$$

is the abelian group of equivalence classes $$[\psi]$$ of non-degenerate ε-quadratic forms $$\psi \in Q_\epsilon(F)$$ 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:


 * $$[\psi] = [\psi'] \Longleftrightarrow n, n' \in {\mathbb N}_0: \psi \oplus H_{(-1)^k}(R)^n \cong \psi' \oplus H_{(-1)^k}(R)^{n'}$$.

The addition in $$L_{2k}(R)$$ is defined by


 * $$[\psi_1] + [\psi_2] := [\psi_1 \oplus \psi_2].$$

The zero element is represented by $$H_{(-1)^k}(R)^n$$ for any $$n \in {\mathbb N}_0$$. The inverse of $$[\psi]$$ is $$[-\psi]$$.

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 $$\pi$$ are the L-groups $$L_*(\mathbf{Z}[\pi])$$ of the group ring $$\mathbf{Z}[\pi]$$. In the applications to topology $$\pi$$ is the fundamental group $$\pi_1 (X)$$ of a space $$X$$. The quadratic L-groups $$L_*(\mathbf{Z}[\pi])$$ 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 $$\mathbf{Z}_2$$ deals with the fixed points of a $$\mathbf{Z}_2$$-action, while the group homology $$H_*$$ deals with the orbits of a $$\mathbf{Z}_2$$-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 $$L_n(R) \to L^n(R)$$ 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 $$L$$-groups $$L_*(\mathbf{Z}[\pi])$$. For finite $$\pi$$ algebraic methods are used, and mostly geometric methods (e.g. controlled topology) are used for infinite $$\pi$$.

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 $$L(e) := L(\mathbf{Z}[e]) = L(\mathbf{Z})$$ 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:
 * $$\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}$$ 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:
 * $$\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}$$ 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.