In
mathematics, topological -theory is a branch of
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 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 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 ...
s, by means of ideas now recognised as (general)
K-theory
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 geom ...
that were introduced by
Alexander Grothendieck. The early work on topological -theory is due to
Michael Atiyah
Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded t ...
and
Friedrich Hirzebruch
Friedrich Ernst Peter Hirzebruch ForMemRS (17 October 1927 – 27 May 2012) was a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation. He has been described a ...
.
Definitions
Let be a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...
and
or
. Then
is defined to be the
Grothendieck group
In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a homomorphic ...
of the
commutative monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids a ...
of
isomorphism class
In mathematics, an isomorphism class is a collection of mathematical objects isomorphic to each other.
Isomorphism classes are often defined as the exact identity of the elements of the set is considered irrelevant, and the properties of the st ...
es of finite-dimensional -vector bundles over under
Whitney sum
In mathematics, a vector bundle is a topology, topological construction that makes precise the idea of a family of vector spaces parameterized by another space (mathematics), space X (for example X could be a topological space, a manifold, or an ...
.
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
In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as x_0, that remains ...
(cf.
reduced homology
In mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, motivated by the intuition that all of the homology groups of a single point should be equal to zero. This modification allows more concise sta ...
). This reduced theory is intuitively modulo
trivial bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a ...
s. 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
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
as follows. The
short exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the conte ...
of a pair of pointed spaces
:
extends to a
long exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the conte ...
:
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
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of ...
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
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, and ...
from the
homotopy category In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed b ...
of (pointed) spaces to the category of commutative rings. Thus, for instance, the -theory over
contractible space
In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within ...
s 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 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 i ...
s:
Similarly,
For real -theory use .
* There is a
natural
Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are ...
ring homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is:
:addition prese ...
the
Chern character
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Ya ...
, 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 In mathematics, the splitting principle is a technique used to reduce questions about vector bundles to the case of line bundles.
In the theory of vector bundles, one often wishes to simplify computations, say of Chern classes. Often computation ...
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 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 paracompact ...
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
In mathematics, operator K-theory is a noncommutative analogue of topological K-theory for Banach algebras with most applications used for C*-algebras.
Overview
Operator K-theory resembles topological K-theory more than algebraic K-theory. In par ...
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
Raoul Bott (September 24, 1923 – December 20, 2005) was a Hungarian- American mathematician known for numerous basic contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott functions whi ...
(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 In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of k- dimensional subspaces of V, given a point in the Grassmannian corresponding to a k-dimensional vector ...
on
i.e. the
Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex number ...
.
*
*
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
John Frank Adams (5 November 1930 – 7 January 1989) was a British mathematician, one of the major contributors to homotopy theory.
Life
He was born in Woolwich, a suburb in south-east London, and attended Bedford School. He began research ...
. First he solved the
Hopf invariant
In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between n-spheres.
__TOC__ Motivation
In 1931 Heinz Hopf used Clifford parallels to construct the '' Hopf map''
:\eta\colon S^3 \t ...
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
Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded t ...
and
Friedrich Hirzebruch
Friedrich Ernst Peter Hirzebruch ForMemRS (17 October 1927 – 27 May 2012) was a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation. He has been described a ...
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 In mathematics, the Atiyah–Hirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by in the special case of topological K-theory. For a CW complex X and a generalized cohomology theory E^\bullet ...
(computational tool for finding K-theory groups)
*
KR-theory
*
Atiyah–Singer index theorem
In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the spac ...
*
Snaith's theorem
*
Algebraic K-theory
Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense ...
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