Topological K-theory

Short description: Branch of algebraic topology

In mathematics, topological K-theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, 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]\displaystyle{ k= \R }[/math] or [math]\displaystyle{ \Complex }[/math]. Then [math]\displaystyle{ K_k(X) }[/math] is defined to be the Grothendieck group of the commutative monoid of isomorphism classes of finite-dimensional k-vector bundles over X under Whitney sum. Tensor product of bundles gives K-theory a commutative ring structure. Without subscripts, [math]\displaystyle{ K(X) }[/math] usually denotes complex K-theory whereas real K-theory is sometimes written as [math]\displaystyle{ KO(X) }[/math]. The remaining discussion is focused on complex K-theory.

As a first example, note that the K-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 K-theory, [math]\displaystyle{ \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]\displaystyle{ \varepsilon_1 }[/math] and [math]\displaystyle{ \varepsilon_2 }[/math], so that [math]\displaystyle{ E \oplus \varepsilon_1 \cong F\oplus \varepsilon_2 }[/math]. 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, [math]\displaystyle{ \widetilde{K}(X) }[/math] can be defined as the kernel of the map [math]\displaystyle{ K(X)\to K(x_0) \cong \Z }[/math] induced by the inclusion of the base point x₀ into X.

K-theory forms a multiplicative (generalized) cohomology theory as follows. The short exact sequence of a pair of pointed spaces (X, A)

[math]\displaystyle{ \widetilde{K}(X/A) \to \widetilde{K}(X) \to \widetilde{K}(A) }[/math]

extends to a long exact sequence

[math]\displaystyle{ \cdots \to \widetilde{K}(SX) \to \widetilde{K}(SA) \to \widetilde{K}(X/A) \to \widetilde{K}(X) \to \widetilde{K}(A). }[/math]

Let Sⁿ be the n-th reduced suspension of a space and then define

[math]\displaystyle{ \widetilde{K}^{-n}(X):=\widetilde{K}(S^nX), \qquad n\geq 0. }[/math]

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:

[math]\displaystyle{ K^{-n}(X)=\widetilde{K}^{-n}(X_+). }[/math]

Here [math]\displaystyle{ X_+ }[/math] is [math]\displaystyle{ X }[/math] with a disjoint basepoint labeled '+' adjoined.

Finally, the Bott periodicity theorem as formulated below extends the theories to positive integers.

Properties

• [math]\displaystyle{ K^n }[/math] (respectively, [math]\displaystyle{ \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 spaces is always [math]\displaystyle{ \Z. }[/math]
• The spectrum of K-theory is [math]\displaystyle{ BU\times\Z }[/math] (with the discrete topology on [math]\displaystyle{ \Z }[/math]), i.e. [math]\displaystyle{ K(X) \cong \left [ X^+, \Z \times BU \right ], }[/math] where [ , ] denotes pointed homotopy classes and BU is the colimit of the classifying spaces of the unitary groups: [math]\displaystyle{ BU(n) \cong \operatorname{Gr} \left (n, \Complex^{\infty} \right ). }[/math] Similarly, [math]\displaystyle{ \widetilde{K}(X) \cong [X, \Z \times BU]. }[/math] For real K-theory use BO.
• There is a natural ring homomorphism [math]\displaystyle{ K^0(X) \to H^{2*}(X, \Q), }[/math] the Chern character, such that [math]\displaystyle{ K^0(X) \otimes \Q \to H^{2*}(X, \Q) }[/math] is an isomorphism.
• The equivalent of the Steenrod operations in K-theory are the Adams operations. 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]\displaystyle{ K(X)\cong\widetilde{K}(T(E)), }[/math] where T(E) is the Thom space of the vector bundle E over X. This holds whenever E is a spin-bundle.
• 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 periodicity named after Raoul Bott (see Bott periodicity theorem) can be formulated this way:

• [math]\displaystyle{ K(X \times \mathbb{S}^2) = K(X) \otimes K(\mathbb{S}^2), }[/math] and [math]\displaystyle{ K(\mathbb{S}^2) = \Z[H]/(H-1)^2 }[/math] where H is the class of the tautological bundle on [math]\displaystyle{ \mathbb{S}^2 = \mathbb{P}^1(\Complex), }[/math] i.e. the Riemann sphere.
• [math]\displaystyle{ \widetilde{K}^{n+2}(X)=\widetilde{K}^n(X). }[/math]
• [math]\displaystyle{ \Omega^2 BU \cong BU \times \Z. }[/math]

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 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.

Chern character

Michael Atiyah and Friedrich Hirzebruch proved a theorem relating the topological K-theory of a finite CW complex [math]\displaystyle{ X }[/math] with its rational cohomology. In particular, they showed that there exists a homomorphism

[math]\displaystyle{ ch : K^*_{\text{top}}(X)\otimes\Q \to H^*(X;\Q) }[/math]

such that

[math]\displaystyle{ \begin{align} K^0_{\text{top}}(X)\otimes \Q & \cong \bigoplus_k H^{2k}(X;\Q) \\ K^1_{\text{top}}(X)\otimes \Q & \cong \bigoplus_k H^{2k+1}(X;\Q) \end{align} }[/math]

There is an algebraic analogue relating the Grothendieck group of coherent sheaves and the Chow ring of a smooth projective variety [math]\displaystyle{ X }[/math].

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

• Atiyah, Michael Francis (1989). K-theory. Advanced Book Classics (2nd ed.). Addison-Wesley. ISBN 978-0-201-09394-0. 
• Friedlander, Eric; Grayson, Daniel, eds (2005). Handbook of K-Theory. Berlin, New York: Springer-Verlag. doi:10.1007/978-3-540-27855-9. ISBN 978-3-540-30436-4. 
• Karoubi, Max (1978). K-theory: an introduction. Classics in Mathematics. Springer-Verlag. doi:10.1007/978-3-540-79890-3. ISBN 0-387-08090-2. 
• Karoubi, Max (2006). "K-theory. An elementary introduction". arXiv:math/0602082.
• Hatcher, Allen (2003). "Vector Bundles & K-Theory". http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html. 
• Stykow, Maxim (2013). "Connections of K-Theory to Geometry and Topology". https://www.researchgate.net/publication/330505308.