In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra 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-functors 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 where it has been conjectured that they classify D-branes, Ramond–Ramond field strengths and also certain spinors on generalized complex manifolds. In condensed matter physics K-theory has been used to classify topological insulators, superconductors and stable Fermi surfaces. For more details, see K-theory (physics).
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 classes of the abelian monoid . Here we will denote the identity element of by so that will be the identity element of First, for any since we can set and apply the equation from the equivalence relation to get This implies
hence we have an additive inverse for each element in . This should give us the hint that we should be thinking of the equivalence classes as formal differences Another useful observation is the invariance of equivalence classes under scaling:
for any
The Grothendieck completion can be viewed as a functor and it has the property that it is left adjoint to the corresponding forgetful functor That means that, given a morphism of an abelian monoid to the underlying abelian monoid of an abelian group there exists a unique abelian group morphism
Example for natural numbers
An illustrative example to look at is the Grothendieck completion of . We can see that For any pair we can find a minimal representative by using the invariance under scaling. For example, we can see from the scaling invariance that
In general, if then
which is of the form or
This shows that we should think of the as positive integers and the as negative integers.
Definitions
There are a number of basic definitions of K-theory: two coming from topology and two from algebraic geometry.
Grothendieck group for compact Hausdorff spaces
Given a compact Hausdorff space consider the set of isomorphism classes of finite-dimensional vector bundles over , denoted and let the isomorphism class of a vector bundle be denoted . Since isomorphism classes of vector bundles behave well with respect to direct sums, we can write these operations on isomorphism classes by
It should be clear that is an abelian monoid where the unit is given by the trivial vector bundle . We can then apply the Grothendieck completion to get an abelian group from this abelian monoid. This is called the K-theory of and is denoted .