In mathematics algebraic Ltheory is the Ktheory of quadratic forms; the term was coined by C. T. C. Wall, with L being used as the letter after K. Algebraic Ltheory, also known as 'hermitian Ktheory', is important in surgery theory.
Contents

Definition 1

Even dimension 1.1

Odd dimension 1.2

Examples and applications 2

References 3
Definition
One can define Lgroups for any ring with involution R: the quadratic Lgroups L_*(R) (Wall) and the symmetric Lgroups L^*(R) (Mishchenko, Ranicki).
Even dimension
The evendimensional Lgroups L_{2k}(R) are defined as the Witt groups of εquadratic forms over the ring R with \epsilon = (1)^k. More precisely,
L_{2k}(R)
is the abelian group of equivalence classes [\psi] of nondegenerate εquadratic forms \psi \in Q_\epsilon(F) over R, where the underlying Rmodules F are finitely generated free. The equivalence relation is given by stabilization with respect to hyperbolic εquadratic 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 odddimensional Lgroups is more complicated; further details and the definition of the odddimensional Lgroups can be found in the references mentioned below.
Examples and applications
The Lgroups of a group \pi are the Lgroups 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 Lgroups L_*(\mathbf{Z}[\pi]) play a central role in the surgery classification of the homotopy types of ndimensional manifolds of dimension n > 4, and in the formulation of the Novikov conjecture.
The distinction between symmetric Lgroups and quadratic Lgroups, 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}_2action, while the group homology H_* deals with the orbits of a \mathbf{Z}_2action; compare X^G (fixed points) and X_G = X/G (orbits, quotient) for upper/lower index notation.
The quadratic Lgroups: L_n(R) and the symmetric Lgroups: L^n(R) are related by a symmetrization map L_n(R) \to L^n(R) which is an isomorphism modulo 2torsion, and which corresponds to the polarization identities.
The quadratic Lgroups are 4fold periodic. Symmetric Lgroups are not 4periodic 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 Lgroups 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 Lgroups for any additive category with a chain duality, as in Ranicki (section 1).
Integers
The simply connected Lgroups are also the Lgroups of the integers, as L(e) := L(\mathbf{Z}[e]) = L(\mathbf{Z}) for both L = L^* or L_*. For quadratic Lgroups, these are the surgery obstructions to simply connected surgery.
The quadratic Lgroups 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 Lgroups detect the signature; in singly even dimension (4k+2), the Lgroups detect the Arf invariant (topologically the Kervaire invariant).
The symmetric Lgroups 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 Lgroups, as with the quadratic Lgroups, detect the signature; in dimension (4k+1), the Lgroups detect the de Rham invariant.
References

Lück, Wolfgang (2002), "A basic introduction to surgery theory", Topology of highdimensional manifolds, No. 1, 2 (Trieste, 2001) (PDF), ICTP Lect. Notes 9, Abdus Salam Int. Cent. Theoret. Phys., Trieste, pp. 1–224,

Ranicki, A. A. (1992), Algebraic Ltheory and topological manifolds (PDF), Cambridge Tracts in Mathematics 102,

Wall, C. T. C. (1999) [1970], Ranicki, Andrew, ed., Surgery on compact manifolds (PDF), Mathematical Surveys and Monographs 69 (2nd ed.), Providence, R.I.:
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