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{aligned}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{aligned}} 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{aligned}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{aligned}} 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.