Topological K-theory
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158828174
2007-09-18T22:43:53Z
Molinagaray
563431
/* References */
In [[mathematics]], '''topological K-theory''' is a branch of [[algebraic topology]]. It was founded to study [[vector bundle]]s on general [[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 <math>k=\mathbb{R}</math> or <math>k=\mathbb{C}</math>. Then <math>K_{k}(X)</math> is the [[Grothendieck group]] of the [[commutative monoid]] whose elements are the [[isomorphism class]]es of finite dimensional <math>k</math>-vector bundles on ''X'' with the operation
:<math>[E]\oplus [F] = [E\oplus F]</math>
for vector bundles ''E'', ''F''. Usually, <math>K_k(X)</math> is denoted <math>KO(X)</math> in real case and <math>KU(X)</math> in the complex case.
More explicitly, '''stable equivalence''', the [[equivalence relation]] on bundles ''E'' and ''F'' on ''X'' of defining the same element in ''K''(''X''), occurs when there is a [[trivial bundle]] ''G'', so that
:<math>E\oplus G\cong F\oplus G</math>.
Under the [[tensor product of vector bundles]] ''K''(''X'') then becomes a [[commutative ring]].
The [[rank of a vector bundle]] carries over to the K-group define the homomorphism
:<math>K(X)\to\check{H}^0(X,\mathbb{Z})</math>
where <math>\check{H}^0(X,\mathbb{Z})</math> is the 0-group de [[Čech cohomology]] which is equal to group of locally constant functions with values in <math>\mathbb{Z}</math>.
If X has a [[pointed space|distinguished basepoint]] x<sub>0</sub>, then the reduced K-group (cf. [[reduced homology]]) satisfies
:<math>K(X)\cong\tilde K(X)\oplus K(\{x_0\})</math>
and is defined as either the [[kernel (category theory)|kernel]] of <math>K(X)\to K(\{x_0\})</math> (where <math>\{x_0\}\to X</math> is basepoint inclusion) or the [[cokernel]] of <math>K(\{x_0\})\to K(X)</math> (where <math>X\to\{x_0\}</math> is the constant map).
When X is a [[connected space]], <math>\tilde K(X)\cong\operatorname{Ker}(K(X)\to\check{H}^0(X,\mathbb{Z})=\mathbb{Z})</math>.
The definition of [[functor]] K extends to [[Category (mathematics)|category]] pairs of [[compact space]]s (an object is a pair <math>(X,Y)</math>, <math>X</math> is compact and <math>Y\subset X</math> is closed, a [[morphism]] between <math>(X,Y)</math> and <math>(X',Y')</math> is a continuous map <math>f:X\to X'</math> such that <math>f(Y)\subset Y'</math> )
:<math>K(X,Y):=\tilde{K}(X/Y)</math>
The reduced K-group is given by <math>x_0=\{Y\}</math>.
The definition
:<math> K_{\mathbb{C}}^{n}(X,Y)=\tilde K_{\mathbb{C}}(S^{|n|}(X/Y))</math>
gives the sequence of K-groups for <math>n\in\mathbb{Z}</math>, where ''S'' denotes the [[reduced suspension]].
== Properties ==
* <math>K^n</math> is a [[contravariant functor]].
* The [[classifying space]] of <math>\tilde{K}</math> is <math>BO_k</math>(BO, in real case; BU in complex case), i.e. <math>K_k(X)\cong[X,BO_k]</math>
* The [[classifying space]] of <math>K</math> is <math>\mathbb{Z}\times BO_k</math> (<math>\mathbb{Z}</math> with discrete topology), i.e. <math>K_k(X)\cong[X,\mathbb{Z}\times BO_k]</math>
* There is a [[natural homomorphism|natural]] [[ring homomorphism]] <math>K^*(X)\to H^{2*}(X,\mathbb{Q})</math>, the [[Chern character]], such that <math>K^*(X)\otimes\mathbb{Q}\to H^{2*}(X,\mathbb{Q})</math> is an isomorphism.
* 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 [[periodicity]] named for [[Raoul Bott]] (see [[Bott periodicity theorem]]) can be formulated this way:
* <math>K(X\times S^2)=K(X)\otimes K(S^2),</math> and <math>K(S^2)=\mathbb Z[H]/(H-1)^2;</math> where <math>H</math> is the class of the [[tautological bundle]] on the <math>S^2=\mathbb CP^1</math>, i.e. the [[Riemann sphere]] as [[complex projective line]]
* <math>\tilde K^{n+2}(X)=\tilde K^n(X)</math>
* <math>\Omega^2\mathrm{BU}\simeq\mathrm{BU}\times\mathbf Z</math>.
In [[real K-theory]] there is a similar periodicity, but ''modulo'' 8.
==References==
*M. Karoubi, [http://www.institut.math.jussieu.fr/~karoubi/KBook.html K-theory, an introduction], 1978 - Berlin; New York: Springer-Verlag
*M.F. Atiyah, D.W. Anderson ''K-Theory'' 1967 - New York, WA Benjamin
[[Category:Algebraic topology]]
[[Category:K-theory]]