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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
DefinitionEdit
One can define L-groups for any ring with involution R: the quadratic L-groups <math>L_*(R)</math> (Wall) and the symmetric L-groups <math>L^*(R)</math> (Mishchenko, Ranicki).
Even dimensionEdit
The even-dimensional L-groups <math>L_{2k}(R)</math> are defined as the Witt groups of ε-quadratic forms over the ring R with <math>\epsilon = (-1)^k</math>. More precisely,
- <math>L_{2k}(R)</math>
is the abelian group of equivalence classes <math>[\psi]</math> of non-degenerate ε-quadratic forms <math>\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>[\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>L_{2k}(R)</math> is defined by
- <math>[\psi_1] + [\psi_2] := [\psi_1 \oplus \psi_2].</math>
The zero element is represented by <math>H_{(-1)^k}(R)^n</math> for any <math>n \in {\mathbb N}_0</math>. The inverse of <math>[\psi]</math> is <math>[-\psi]</math>.
Odd dimensionEdit
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 applicationsEdit
The L-groups of a group <math>\pi</math> are the L-groups <math>L_*(\mathbf{Z}[\pi])</math> of the group ring <math>\mathbf{Z}[\pi]</math>. In the applications to topology <math>\pi</math> is the fundamental group <math>\pi_1 (X)</math> of a space <math>X</math>. The quadratic L-groups <math>L_*(\mathbf{Z}[\pi])</math> play a central role in the surgery classification of the homotopy types of <math>n</math>-dimensional manifolds of dimension <math>n > 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>H^*</math> of the cyclic group <math>\mathbf{Z}_2</math> deals with the fixed points of a <math>\mathbf{Z}_2</math>-action, while the group homology <math>H_*</math> deals with the orbits of a <math>\mathbf{Z}_2</math>-action; compare <math>X^G</math> (fixed points) and <math>X_G = X/G</math> (orbits, quotient) for upper/lower index notation.
The quadratic L-groups: <math>L_n(R)</math> and the symmetric L-groups: <math>L^n(R)</math> are related by a symmetrization map <math>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>L</math>-groups <math>L_*(\mathbf{Z}[\pi])</math>. For finite <math>\pi</math> algebraic methods are used, and mostly geometric methods (e.g. controlled topology) are used for infinite <math>\pi</math>.
More generally, one can define L-groups for any additive category with a chain duality, as in Ranicki (section 1).
IntegersEdit
The simply connected L-groups are also the L-groups of the integers, as <math>L(e) := L(\mathbf{Z}[e]) = L(\mathbf{Z})</math> for both <math>L</math> = <math>L^*</math> or <math>L_*.</math> For quadratic L-groups, these are the surgery obstructions to simply connected surgery.
The quadratic L-groups of the integers are:
- <math>\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>\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.