L-theory

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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[edit]

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[edit]

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[edit]

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[edit]

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 L-groups are 4-fold periodic. Symmetric L-groups are not 4-periodic in general (see Ranicki, page 12), though they are for the integers.

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[edit]

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.

References[edit]