In
mathematics, K-theory is, roughly speaking, the study of a
ring generated by
vector bundles
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every ...
over a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
or
scheme. 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 classif ...
, it is a
cohomology theory known as
topological K-theory. In
algebra
Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
and
algebraic geometry, it is referred to as
algebraic K-theory. It is also a fundamental tool in the field of
operator algebras. It can be seen as the study of certain kinds of
invariants of large
matrices.
K-theory involves the construction of families of ''K''-
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, an ...
s that map from topological spaces or schemes to associated rings; these rings reflect some aspects of the structure of the original spaces or schemes. As with functors to
groups in algebraic topology, the reason for this functorial mapping is that it is easier to compute some topological properties from the mapped rings than from the original spaces or schemes. Examples of results gleaned from the K-theory approach include the
Grothendieck–Riemann–Roch theorem,
Bott periodicity, the
Atiyah–Singer index theorem, and the
Adams operations.
In
high energy physics, K-theory and in particular
twisted K-theory have appeared in
Type II string theory
In theoretical physics, type II string theory is a unified term that includes both type IIA strings and type IIB strings theories. Type II string theory accounts for two of the five consistent superstring theories in ten dimensions. Both theories ...
where it has been conjectured that they classify
D-branes
In string theory, D-branes, short for ''Dirichlet membrane'', are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Jin Dai, Leigh, and Polchi ...
,
Ramond–Ramond field strengths and also certain
spinors on
generalized complex manifolds. In
condensed matter physics
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the s ...
K-theory has been used to classify
topological insulators,
superconductors and stable
Fermi surfaces. For more details, see
K-theory (physics)
In string theory, K-theory classification refers to a conjectured application of K-theory (in abstract algebra and algebraic topology) to superstrings, to classify the allowed Ramond–Ramond field strengths as well as the charges of stable D-bra ...
.
Grothendieck completion
The Grothendieck completion of an
abelian monoid into an abelian group is a necessary ingredient for defining K-theory since all definitions start by constructing an abelian monoid from a suitable category and turning it into an abelian group through this universal construction. Given an abelian monoid
let
be the relation on
defined by
:
if there exists a
such that
Then, the set
has the structure of a
group where:
:
Equivalence classes in this group should be thought of as formal differences of elements in the abelian monoid. This group
is also associated with a monoid homomorphism
given by
which has a
certain universal property.
To get a better understanding of this group, consider some
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
es of the abelian monoid
. Here we will denote the identity element of
by
so that