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

Even dimension

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

${\displaystyle L_{2k}(R)}$

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

${\displaystyle [\psi ]=[\psi ']\Longleftrightarrow n,n'\in {\mathbb {N} }_{0}:\psi \oplus H_{(-1)^{k}}(R)^{n}\cong \psi '\oplus H_{(-1)^{k}}(R)^{n'}}$.

The addition in ${\displaystyle L_{2k}(R)}$ is defined by

${\displaystyle [\psi _{1}]+[\psi _{2}]:=[\psi _{1}\oplus \psi _{2}].}$

The zero element is represented by ${\displaystyle H_{(-1)^{k}}(R)^{n}}$ for any ${\displaystyle n\in {\mathbb {N} }_{0}}$. The inverse of ${\displaystyle [\psi ]}$ is ${\displaystyle [-\psi ]}$.

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 ${\displaystyle \pi }$ are the L-groups ${\displaystyle L_{*}(\mathbf {Z} [\pi ])}$ of the group ring ${\displaystyle \mathbf {Z} [\pi ]}$. In the applications to topology ${\displaystyle \pi }$ is the fundamental group ${\displaystyle \pi _{1}(X)}$ of a space ${\displaystyle X}$. The quadratic L-groups ${\displaystyle L_{*}(\mathbf {Z} [\pi ])}$ play a central role in the surgery classification of the homotopy types of ${\displaystyle n}$-dimensional manifolds of dimension ${\displaystyle n>4}$, 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 ${\displaystyle H^{*}}$ of the cyclic group ${\displaystyle \mathbf {Z} _{2}}$ deals with the fixed points of a ${\displaystyle \mathbf {Z} _{2}}$-action, while the group homology ${\displaystyle H_{*}}$ deals with the orbits of a ${\displaystyle \mathbf {Z} _{2}}$-action; compare ${\displaystyle X^{G}}$ (fixed points) and ${\displaystyle X_{G}=X/G}$ (orbits, quotient) for upper/lower index notation.

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

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

The quadratic L-groups of the integers are:

{\displaystyle {\begin{aligned}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{aligned}}}

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:

{\displaystyle {\begin{aligned}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{aligned}}}

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) (PDF), ICTP Lect. Notes, 9, Abdus Salam Int. Cent. Theoret. Phys., Trieste, pp. 1–224, MR 1937016
• Ranicki, Andrew A. (1992), Algebraic L-theory and topological manifolds (PDF), Cambridge Tracts in Mathematics, 102, Cambridge University Press, ISBN 978-0-521-42024-2, MR 1211640
• Wall, C. T. C. (1999) [1970], Ranicki, Andrew (ed.), Surgery on compact manifolds (PDF), Mathematical Surveys and Monographs, 69 (2nd ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0942-6, MR 1687388