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In [[mathematics]], '''topological ''K''-theory''' is a branch of [[algebraic topology]]. It was founded to study [[vector bundle]]s on [[topological space]]s, by means of ideas now recognised as (general) [[K-theory]] that were introduced by [[Alexander Grothendieck]]. The early work on topological ''K''-theory is due to [[Michael Atiyah]] and [[Friedrich Hirzebruch]].

== Definitions ==
Let ''X'' be a [[compact Hausdorff space]] and ''k''='''R''', '''C'''. Then ''K<sub>k</sub>''(''X'') is the [[Grothendieck group]] of the [[commutative monoid]] of [[isomorphism class]]es of finite dimensional ''k''-vector bundles over ''X'' under [[Whitney sum]]. [[Whitney sum#Operations on vector bundles|Tensor product]] of bundles gives ''K''-theory a [[commutative ring]] structure. Without subscripts, ''K''(''X'') usually denotes complex ''K''-theory whereas real ''K''-theory is sometimes written as ''KO''(''X''). The remaining discussion is focussed on complex ''K''-theory, the real case being similar.

As a first example, note that the ''K''-theory of a point are 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 are the integers.

There is also a reduced version of ''K''-theory, <math>\widetilde{K}(X)</math>, defined for ''X'' a compact [[pointed space]] (cf. [[reduced homology]]). This reduced theory is intuitively ''K''(''X'') modulo trivial bundles. It is defined as the group of stable equivalence classes of bundles. Two bundles ''E'' and ''F'' are said to be '''stably isomorphic''' if there are trivial bundles <math>\epsilon_1</math> and <math>\epsilon_2</math>, so that <math>E\oplus\epsilon_1\cong F\oplus\epsilon_2</math>. The fact that this equivalence relation results in a group follows from the fact that every vector bundle can be completed to a trivial bundle by summing with its orthogonal complement. Alternatively, <math>\widetilde{K}(X)</math> can be defined as the kernel of the map <math>K(X)\to K(x_0)\cong\mathbb{Z}</math> induced by the inclusion of the basepoint ''x<sub>0</sub>'' into ''X''.

''K''-theory forms a multiplicative (generalized) [[Cohomology#Cohomology theories|cohomology theory]] as follows. The short exact sequence of a pair of pointed spaces (''X'',''A'')

<math>\widetilde{K}(X/A)\to\widetilde{K}(X)\to\widetilde{K}(A)</math>

extends to a long exact sequence

<math>\cdots\to\widetilde{K}(SX)\to\widetilde{K}(SA)\to\widetilde{K}(X/A)\to\widetilde{K}(X)\to\widetilde{K}(A)</math>.

Then define <math>\widetilde{K}^{-n}(X):=\widetilde{K}(S^nX)</math> for <math>n\geq 0</math> where <math>S^n</math> is the ''n''th [[Suspension (topology)|reduced suspension]] of a space. Negative indices are chosen so that the coboundary maps increase dimension. [[One-point compactification]] extends this definition to locally compact spaces without basepoints: <math>K^{-n}(X)=\widetilde{K}^{-n}(X_+)</math>. Finally, the Bott periodicity theorem as formulated below extends the theories to positive integers.

== Properties ==
* <math>K^n</math> respectively <math>\widetilde{K}^n</math> is a [[contravariant functor]] from the [[homotopy category]] of (pointed) spaces to the category of commutative rings. Thus, for instance, the ''K''-theory over [[contractible space]]s is always '''Z'''.
* The [[Spectrum (homotopy theory)|spectrum]] of ''K''-theory is ''BU'' × '''Z''' ('''Z''' with the discrete topology), i.e. <math>K(X)\cong[X_+,\mathbb{Z}\times BU]</math> where [,] denotes pointed homotopy classes and ''BU'' is the [[colimit]] of the classifying spaces <math>BU_n\cong Gr_n(\mathbb{C}^{\infty})</math> of the [[unitary group]]s. Similarly, <math>\widetilde{K}(X)\cong[X,\mathbb{Z}\times BU]</math>. For real ''K''-theory use ''BO''.
* There is a [[natural homomorphism|natural]] [[ring homomorphism]] <math>K^*(X)\to H^{2*}(X,\mathbf{Q})</math>, the [[Chern character]], such that <math>K^*(X)\otimes\mathbf{Q}\to H^{2*}(X,\mathbf{Q})</math> is an isomorphism.
* The equivalent of the [[Steenrod operation]]s in ''K''-theory are the [[Adams operation]]s. They can be used to define characteristic classes in topological ''K''-theory.
* The [[Splitting principle]] of topological ''K''-theory allows one to reduce statements about arbitrary vector bundles to statements about sums of line bundles.
* The [[Thom isomorphism theorem]] in topological ''K''-theory is <math>K(X)\cong\widetilde{K}(T(E))</math> where ''T(E)'' is the [[Thom space]] of the vector bundle ''E'' over ''X''.
* The [[Atiyah-Hirzebruch spectral sequence]] allows computation of ''K''-groups from ordinary cohomology groups.
* Topological ''K''-theory can be generalized vastly to a functor on [[C*-algebras]], see [[operator K-theory]] and [[KK-theory]].

== Bott periodicity ==
The phenomenon of [[Frequency|periodicity]] named after [[Raoul Bott]] (see [[Bott periodicity theorem]]) can be formulated this way:

* ''K''(''X'' × '''S'''<sup>2</sup>)= ''K''(''X'') ⊗ ''K''('''S'''<sup>2</sup>), and ''K''('''S'''<sup>2</sup>) = '''Z'''[''H'']/(''H'' - 1)<sup>2</sup> where ''H'' is the class of the [[tautological bundle]] on '''S'''<sup>2</sup> = '''CP'''<sup>1</sup>, i.e. the [[Riemann sphere]].
* <math>\widetilde{K}^{n+2}(X)=\widetilde{K}^n(X).</math>
* Ω<sup>2</sup>BU ≅ BU &times; '''Z'''.

In real ''K''-theory there is a similar periodicity, but modulo 8.

== Applications ==

The two most famous applications of topological ''K''-theory are both due to [[J. F. Adams]]. First he solved the [[Hopf invariant]] one problem by doing a computation with his [[Adams operation]]s. Then he proved an upper bound for the number of linearly independent [[vector fields on spheres]].

==See also==
*[[KR-theory]]
*[[Atiyah-Singer index theorem]]
*[[Snaith’s theorem]]

==References==
* {{Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | title=K-theory | publisher=[[Addison-Wesley]] | edition=2nd | series=Advanced Book Classics | isbn=978-0-201-09394-0 | mr=1043170 | year=1989}}
*{{Citation | editor1-last=Friedlander | editor1-first=Eric | editor2-last=Grayson | editor2-first=Daniel | title=Handbook of K-Theory | url=http://www.springerlink.com/content/978-3-540-23019-9/ | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-30436-4 | mr=2182598 | year=2005}}
* [[Max Karoubi]] (1978), [http://www.institut.math.jussieu.fr/~karoubi/KBook.html K-theory, an introduction] Springer-Verlag
** Max Karoubi (2006), "K-theory. An elementary introduction", {{arxiv|math|0602082}}
* [[Allen Hatcher]], ''[http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html Vector Bundles & K-Theory]'', (2003)
* Maxim Stykow, ''[http://www.math.ubc.ca/~maxim/K-Theory.pdf Connections of K-Theory to Geometry and Topology]'', (2013)

[[Category:Algebraic topology]]
[[Category:K-theory]]