In
mathematics, a direct limit of groups is the
direct limit of a of
groups
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
. These are central objects of study in
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, especially
stable homotopy theory and
homological algebra. They are sometimes called stable groups, though this term normally means something quite different in
model theory.
Certain examples of stable groups are easier to study than "unstable" groups, the groups occurring in the limit. This is surprising, given that they are generally infinite-dimensional, constructed as limits of groups with finite-dimensional representations.
Examples
Each family of
classical group
In mathematics, the classical groups are defined as the special linear groups over the reals , the complex numbers and the quaternions together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or s ...
s forms a direct system, via inclusion of
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
in the upper left corner, such as
. The stable groups are denoted
or
.
Bott periodicity
In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by , which proved to be of foundational significance for much further research, in particular in K-theory of stable comp ...
computes the homotopy of the stable
unitary group
In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is ...
and stable
orthogonal group.
The
Whitehead group of a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
(the
first K-group) can be defined in terms of
.
Stable homotopy groups of spheres are the stable groups associated with the
suspension functor.
See also
*
*
*
References
{{reflist
Homotopy theory
Homological algebra
Algebraic topology