Algebraic
L-theory is the
K-theoryIn mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...
of
quadratic formIn mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
s; the term was coined by
C. T. C. WallCharles Terence Clegg Wall is a leading British mathematician, educated at Marlborough and Trinity College, Cambridge. He is an emeritus professor of the University of Liverpool, where he was first appointed Professor in 1965...
,
with
L being used as the letter after
K. Algebraic
L-theory, also known as 'hermitian
K-theory',
is important in
surgery theoryIn mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one manifold from another in a 'controlled' way, introduced by . Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along...
.
Definition
One can define
L-groups for any ring with involution
R: the quadratic
L-groups
(Wall) and the symmetric
L-groups
(Mishchenko, Ranicki).
Even dimension
The even dimensional
L-groups
are defined as the
Witt groupIn mathematics, a Witt group of a field, named after Ernst Witt, is an abelian group whose elements are represented by symmetric bilinear forms over the field.-Definition:Fix a field k. All vector spaces will be assumed to be finite-dimensional...
s of ε-quadratic forms over the ring
R with
. More precisely,
is the abelian group of equivalence classes
of non-degenerate ε-quadratic forms
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:
.
The addition in
is defined by
The zero element is represented by
for any
. The inverse of
is
.
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
are the
L-groups
of the
group ringIn algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring and its basis is one-to-one with the given group. As a ring, its addition law is that of the free...
. In the applications to topology
is the
fundamental groupIn mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
of a space
. The quadratic
L-groups
play a central role in the surgery classification of the homotopy types of
-dimensional manifolds of dimension
, and in the formulation of the
Novikov conjectureThe Novikov conjecture is one of the most important unsolved problems in topology. It is named for Sergei Novikov who originally posed the conjecture in 1965....
.
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 cohomologyIn abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. The study of fixed points of groups acting on modules and quotient modules...
of the cyclic group
deals with the fixed points of a
-action, while the group homology
deals with the orbits of a
-action; compare
(fixed points) and
(orbits, quotient) for upper/lower index notation.
The quadratic
L-groups:
and the symmetric
L-groups:
are related by
a symmetrization map
which is an isomorphism modulo 2-torsion, and which corresponds to the polarization identities.
The quadratic
L-groups are 4-fold periodic. Symmetric
L-groups are not 4-periodic in general (see Ranicki, page 12), though they are for the integers.
In view of the applications to the
classification of manifoldsIn mathematics, specifically geometry and topology, the classification of manifolds is a basic question, about which much is known, and many open questions remain.-Overview:...
there are extensive calculations of
the quadratic
-groups
. For finite
algebraic methods are used, and mostly geometric methods (e.g. controlled topology) are used for infinite
.
More generally, one can define
L-groups for any
additive categoryIn mathematics, specifically in category theory, an additive category is a preadditive category C such that all finite collections of objects A1,...,An of C have a biproduct A1 ⊕ ⋯ ⊕ An in C....
with a
chain duality, as in Ranicki (section 1).
Integers
The
simply connected L-groups are also the
L-groups of the integers, as
for both
=
or
For quadratic
L-groups, these are the surgery obstructions to simply connected surgery.
The quadratic
L-groups of the integers are:
In doubly even dimension (4
k), the quadratic
L-groups detect the
signatureIn the mathematical field of topology, the signature is an integer invariant which is defined for an oriented manifold M of dimension d=4k divisible by four ....
; in singly even dimension (4
k+2), the
L-groups detect the
Arf invariantIn mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by when he started the systematic study of quadratic forms over arbitrary fields of characteristic2. The Arf invariant is the substitute, in...
.
The symmetric
L-groups of the integers are:
In doubly even dimension (4
k), the symmetric
L-groups, as with the quadratic
L-groups, detect the signature; in dimension (4
k+1), the
L-groups detect the
de Rham invariant.