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 $ϵ=(-1{)}^k$. More precisely,

$L_{2k}(R)$

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

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

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 $L_{*}(𝐙[\pi ])$ of the group ring $𝐙[\pi ]$. In the applications to topology $\pi$ is the fundamental group ${\pi }_1(X)$ of a space $X$. The quadratic L-groups $L_{*}(𝐙[\pi ])$ 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 ${𝐙}_2$ deals with the fixed points of a ${𝐙}_2$-action, while the group homology $H_{*}$ deals with the orbits of a ${𝐙}_2$-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 $L_{*}(𝐙[\pi ])$. 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 $L(e):=L(𝐙[e])=L(𝐙)$ 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:

$\begin{array}{cccc}{L}_{4k}(𝐙)& =𝐙& & \text{signature}/8\\ L_{4k+1}(𝐙)& =0\\ L_{4k+2}(𝐙)& =𝐙/2& & \text{Arf invariant}\\ L_{4k+3}(𝐙)& =0.\end{array}$

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:

$\begin{array}{cccc}{L}^{4k}(𝐙)& =𝐙& & \text{signature}\\ L^{4k+1}(𝐙)& =𝐙/2& & \text{de Rham invariant}\\ L^{4k+2}(𝐙)& =0\\ L^{4k+3}(𝐙)& =0.\end{array}$

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.

References

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