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 [math]\displaystyle{ L_*(R) }[/math] (Wall) and the symmetric L-groups [math]\displaystyle{ L^*(R) }[/math] (Mishchenko, Ranicki).

Even dimension

The even-dimensional L-groups [math]\displaystyle{ L_{2k}(R) }[/math] are defined as the Witt groups of ε-quadratic forms over the ring R with [math]\displaystyle{ \epsilon = (-1)^k }[/math]. More precisely,

[math]\displaystyle{ L_{2k}(R) }[/math]

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

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

The addition in [math]\displaystyle{ L_{2k}(R) }[/math] is defined by

[math]\displaystyle{ [\psi_1] + [\psi_2] := [\psi_1 \oplus \psi_2]. }[/math]

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

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 [math]\displaystyle{ \pi }[/math] are the L-groups [math]\displaystyle{ L_*(\mathbf{Z}[\pi]) }[/math] of the group ring [math]\displaystyle{ \mathbf{Z}[\pi] }[/math]. In the applications to topology [math]\displaystyle{ \pi }[/math] is the fundamental group [math]\displaystyle{ \pi_1 (X) }[/math] of a space [math]\displaystyle{ X }[/math]. The quadratic L-groups [math]\displaystyle{ L_*(\mathbf{Z}[\pi]) }[/math] play a central role in the surgery classification of the homotopy types of [math]\displaystyle{ n }[/math]-dimensional manifolds of dimension [math]\displaystyle{ n \gt 4 }[/math], 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 [math]\displaystyle{ H^* }[/math] of the cyclic group [math]\displaystyle{ \mathbf{Z}_2 }[/math] deals with the fixed points of a [math]\displaystyle{ \mathbf{Z}_2 }[/math]-action, while the group homology [math]\displaystyle{ H_* }[/math] deals with the orbits of a [math]\displaystyle{ \mathbf{Z}_2 }[/math]-action; compare [math]\displaystyle{ X^G }[/math] (fixed points) and [math]\displaystyle{ X_G = X/G }[/math] (orbits, quotient) for upper/lower index notation.

The quadratic L-groups: [math]\displaystyle{ L_n(R) }[/math] and the symmetric L-groups: [math]\displaystyle{ L^n(R) }[/math] are related by a symmetrization map [math]\displaystyle{ L_n(R) \to L^n(R) }[/math] 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 [math]\displaystyle{ L }[/math]-groups [math]\displaystyle{ L_*(\mathbf{Z}[\pi]) }[/math]. For finite [math]\displaystyle{ \pi }[/math] algebraic methods are used, and mostly geometric methods (e.g. controlled topology) are used for infinite [math]\displaystyle{ \pi }[/math].

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 [math]\displaystyle{ L(e) := L(\mathbf{Z}[e]) = L(\mathbf{Z}) }[/math] for both [math]\displaystyle{ L }[/math] = [math]\displaystyle{ L^* }[/math] or [math]\displaystyle{ L_*. }[/math] For quadratic L-groups, these are the surgery obstructions to simply connected surgery.

The quadratic L-groups of the integers are:

[math]\displaystyle{ \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} }[/math]

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:

[math]\displaystyle{ \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} }[/math]

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

• Lück, Wolfgang (2002), "A basic introduction to surgery theory", Topology of high-dimensional manifolds, No. 1, 2 (Trieste, 2001), ICTP Lect. Notes, 9, Abdus Salam Int. Cent. Theoret. Phys., Trieste, pp. 1–224, https://www.him.uni-bonn.de/lueck/data/ictp.pdf 
• Ranicki, Andrew A. (1992), Algebraic L-theory and topological manifolds, Cambridge Tracts in Mathematics, 102, Cambridge University Press, ISBN 978-0-521-42024-2, http://www.maths.ed.ac.uk/~aar/books/topman.pdf 
• Wall, C. T. C. (1999), Ranicki, Andrew, ed., Surgery on compact manifolds, Mathematical Surveys and Monographs, 69 (2nd ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0942-6, http://www.maths.ed.ac.uk/~aar/books/scm.pdf