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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 classify ...
. 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 every p ...
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 poin ...
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 geometr ...
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 th ...
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 as ...
.


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 Britis ...
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 ma ...
and k= \R or \Complex. Then K_k(X) 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 ar ...
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 stru ...
es of finite-dimensional -vector bundles over under
Whitney sum In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
.
Tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same 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 element of V \otime ...
of bundles gives -theory a
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
structure. Without subscripts, K(X) usually denotes complex -theory whereas real -theory is sometimes written as KO(X). 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, \widetilde(X), 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 u ...
(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 stat ...
). 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 p ...
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 \varepsilon_1 and \varepsilon_2, so that E \oplus \varepsilon_1 \cong F\oplus \varepsilon_2. 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, \widetilde(X) 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 learn ...
of the map K(X)\to K(x_0) \cong \Z 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 viewed ...
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 context ...
of a pair of pointed spaces :\widetilde(X/A) \to \widetilde(X) \to \widetilde(A) 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 context ...
:\cdots \to \widetilde(SX) \to \widetilde(SA) \to \widetilde(X/A) \to \widetilde(X) \to \widetilde(A). 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. T ...
of a space and then define :\widetilde^(X):=\widetilde(S^nX), \qquad n\geq 0. 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 th ...
maps increase dimension. It is often useful to have an unreduced version of these groups, simply by defining: :K^(X)=\widetilde^(X_+). Here X_+ is X 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 comp ...
as formulated below extends the theories to positive integers.


Properties

* K^n (respectively, \widetilde^n) 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 be ...
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 that ...
s is always \Z. * 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 colors ...
of -theory is BU\times\Z (with the discrete topology on \Z), i.e. K(X) \cong \left X^+, \Z \times BU \right 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 such ...
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 is ...
s: BU(n) \cong \operatorname \left (n, \Complex^ \right ). Similarly, \widetilde(X) \cong , \Z \times BU 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 ar ...
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 preser ...
K^0(X) \to H^(X, \Q), 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 K^0(X) \otimes \Q \to H^(X, \Q) is an isomorphism. * The equivalent of the Steenrod operations 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 paracompact ...
in topological -theory is K(X)\cong\widetilde(T(E)), 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 s ...
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, 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 pa ...
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 inf ...
.


Bott periodicity

The phenomenon of
periodicity Periodicity or periodic may refer to: Mathematics * Bott periodicity theorem, addresses Bott periodicity: a modulo-8 recurrence relation in the homotopy groups of classical groups * Periodic function, a function whose output contains values tha ...
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 comp ...
) can be formulated this way: * K(X \times \mathbb^2) = K(X) \otimes K(\mathbb^2), and K(\mathbb^2) = \Z (H-1)^2 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-dimension (vector space), dimensional linear subspace, subspaces of V, given a point in the Grassmannian ...
on \mathbb^2 = \mathbb^1(\Complex), 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 numbers ...
. * \widetilde^(X)=\widetilde^n(X). * \Omega^2 BU \cong BU \times \Z. 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 researc ...
. 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 \to S ...
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 ...
.


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 th ...
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 as ...
proved a theorem relating the topological K-theory of a finite CW complex X with its rational cohomology. In particular, they showed that there exists a homomorphism :ch : K^*_(X)\otimes\Q \to H^*(X;\Q) such that :\begin K^0_(X)\otimes \Q & \cong \bigoplus_k H^(X;\Q) \\ K^1_(X)\otimes \Q & \cong \bigoplus_k H^(X;\Q) \end There is an algebraic analogue relating the Grothendieck group of coherent sheaves and the Chow ring of a smooth projective variety X.


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 In mathematics, KR-theory is a variant of topological K-theory defined for spaces with an involution. It was introduced by , motivated by applications to the Atiyah–Singer index theorem for real elliptic operators. Definition A ''real space' ...
*
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 sp ...
*
Snaith's theorem In algebraic topology, a branch of mathematics, Snaith's theorem, introduced by Victor Snaith, identifies the complex K-theory spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of ...
*
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