In
mathematics, topological -theory is a branch of
algebraic topology. It was founded to study
vector bundle
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 ev ...
s on
topological spaces, by means of ideas now recognised as (general)
K-theory that were introduced by
Alexander Grothendieck. The early work on topological -theory is due to
Michael Atiyah and
Friedrich Hirzebruch.
Definitions
Let be a
compact Hausdorff space and
or
. Then
is defined to be the
Grothendieck group of the
commutative monoid of
isomorphism classes of finite-dimensional -vector bundles over under
Whitney sum.
Tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...
of bundles gives -theory a
commutative ring structure. Without subscripts,
usually denotes complex -theory whereas real -theory is sometimes written as
. The remaining discussion is focused on complex -theory.
As a first example, note that the -theory of a point is the integers. This is because vector bundles over a point are trivial and thus classified by their rank and the Grothendieck group of the natural numbers is the integers.
There is also a reduced version of -theory,
, defined for a compact
pointed space (cf.
reduced homology). This reduced theory is intuitively modulo
trivial bundles. It is defined as the group of stable equivalence classes of bundles. Two bundles and are said to be stably isomorphic if there are trivial bundles
and
, so that
. This equivalence relation results in a group since every vector bundle can be completed to a trivial bundle by summing with its orthogonal complement. Alternatively,
can be defined as the
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine lea ...
of the map
induced by the inclusion of the base point into .
-theory forms a multiplicative (generalized)
cohomology theory as follows. The
short exact sequence of a pair of pointed spaces
:
extends to a
long exact sequence
:
Let be the -th
reduced suspension In topology, a branch of mathematics, the suspension of a topological space ''X'' is intuitively obtained by stretching ''X'' into a cylinder and then collapsing both end faces to points. One views ''X'' as "suspended" between these end points. The ...
of a space and then define
:
Negative indices are chosen so that the
coboundary maps increase dimension.
It is often useful to have an unreduced version of these groups, simply by defining:
:
Here
is
with a disjoint basepoint labeled '+' adjoined.
Finally, the
Bott periodicity theorem
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 co ...
as formulated below extends the theories to positive integers.
Properties
*
(respectively,
) is a
contravariant functor from the
homotopy category of (pointed) spaces to the category of commutative rings. Thus, for instance, the -theory over
contractible spaces is always
* The
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of color ...
of -theory is
(with the discrete topology on
), i.e.
where denotes pointed homotopy classes and is the
colimit
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions su ...
of the classifying spaces of the
unitary groups:
Similarly,
For real -theory use .
* There is a
natural ring homomorphism the
Chern character, such that
is an isomorphism.
* The equivalent of the
Steenrod operation In algebraic topology, a Steenrod algebra was defined by to be the algebra of stable cohomology operations for mod p cohomology.
For a given prime number p, the Steenrod algebra A_p is the graded Hopf algebra over the field \mathbb_p of order p ...
s in -theory are the
Adams operations. They can be used to define characteristic classes in topological -theory.
* The
Splitting principle of topological -theory allows one to reduce statements about arbitrary vector bundles to statements about sums of line bundles.
* The
Thom isomorphism theorem In mathematics, the Thom space, Thom complex, or Pontryagin–Thom construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompa ...
in topological -theory is
where is the
Thom space of the vector bundle over . This holds whenever is a spin-bundle.
* The
Atiyah-Hirzebruch spectral sequence allows computation of -groups from ordinary cohomology groups.
* Topological -theory can be generalized vastly to a functor on
C*-algebras
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of cont ...
, see
operator K-theory and
KK-theory
In mathematics, ''KK''-theory is a common generalization both of K-homology and K-theory as an additive bivariant functor on separable C*-algebras. This notion was introduced by the Russian mathematician Gennadi Kasparov in 1980.
It was influ ...
.
Bott periodicity
The phenomenon of
periodicity named after
Raoul Bott (see
Bott periodicity theorem
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 co ...
) can be formulated this way:
*
and
where ''H'' is the class of the
tautological bundle on
i.e. the
Riemann sphere.
*
*
In real -theory there is a similar periodicity, but modulo 8.
Applications
The two most famous applications of topological -theory are both due to
Frank Adams. First he solved the
Hopf invariant one problem by doing a computation with his
Adams operations. Then he proved an upper bound for the number of linearly independent
vector fields on spheres
In mathematics, the discussion of vector fields on spheres was a classical problem of differential topology, beginning with the hairy ball theorem, and early work on the classification of division algebras.
Specifically, the question is how many l ...
.
Chern character
Michael Atiyah and
Friedrich Hirzebruch proved a theorem relating the topological K-theory of a finite CW complex
with its rational cohomology. In particular, they showed that there exists a homomorphism
:
such that
:
There is an algebraic analogue relating the Grothendieck group of coherent sheaves and the Chow ring of a smooth projective variety
.
See also
*
Atiyah–Hirzebruch spectral sequence (computational tool for finding K-theory groups)
*
KR-theory
*
Atiyah–Singer index theorem
*
Snaith's theorem
*
Algebraic K-theory
References
*
*
*
*
*
* {{cite web , last1=Stykow , first1=Maxim , authorlink1=Maxim Stykow , year=2013 , title=Connections of K-Theory to Geometry and Topology , url=https://www.researchgate.net/publication/330505308
K-theory