L-theory explained

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.^[1]

Definition

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

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 { N} _{0:} \psi ⊕ H _{( -1 ) ^{k}} (R) ^{n} \cong \psi' ⊕ H _{( -1 ) ^{k}} (R) ^{n'} .

The addition in

L _{2k} (R) is defined by [\psi _{1]} + [\psi _{2]} : = [\psi _{1} ⊕ \psi _{2].}

The zero element is represented by

H _{( -1 ) ^{k}} (R) ^{n} for any n \in { N} _{0} . The inverse of [\psi] is [ - \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

\pi are the L-groups L _{*(Z[\pi])} of the group ring Z [\pi] . In the applications to topology \pi is the fundamental group\pi _{1} (X) of a space X . The quadratic L-groups L _{*(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.H ^{*} of the cyclic group Z _{2} deals with the fixed points of a Z _{2} -action, while the group homology H _{*} deals with the orbits of a 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 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 ofthe quadratic

L -groups L _{*(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

The simply connected L-groups are also the L-groups of the integers, as

L(e) : = L( Z [e]) = L( 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} ( Z ) & = Z && signature / 8 \\ L _{4k+1} ( Z ) & = 0 \\ L _{4k+2} ( Z ) & = Z / 2 && Arfinvariant \\ L _{4k+3} ( 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} ( Z ) & = Z && signature \\ L ^{4k+1} ( Z ) & = Z / 2 && deRhaminvariant \\ L ^{4k+2} ( Z ) & = 0 \\ L ^{4k+3} ( 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.

Notes and References

1. Web site: L-theory, K-theory and involutions, by Levikov, Filipp, 2013, On University of Aberdeen(ISNI:0000 0004 2745 8820).