Topological K-theory

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 $k=\mathbb {R}$ or $\mathbb {C}$. Then $K_{k}(X)$ 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, $K(X)$ usually denotes complex K-theory whereas real K-theory is sometimes written as $KO(X)$. 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, ${\widetilde {K}}(X)$, 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 $\varepsilon _{1}$ 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, ${\widetilde {K}}(X)$ can be defined as the kernel of the map $K(X)\to K(x_{0})\cong \mathbb {Z}$ 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)

${\widetilde {K}}(X/A)\to {\widetilde {K}}(X)\to {\widetilde {K}}(A)$

extends to a long exact sequence

$\cdots \to {\widetilde {K}}(SX)\to {\widetilde {K}}(SA)\to {\widetilde {K}}(X/A)\to {\widetilde {K}}(X)\to {\widetilde {K}}(A).$

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

${\widetilde {K}}^{-n}(X):={\widetilde {K}}(S^{n}X),\qquad n\geq 0.$

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:

$K^{-n}(X)={\widetilde {K}}^{-n}(X_{+}).$

Here $X_{+}$ is $X$ with a disjoint basepoint labeled '+' adjoined.

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

Properties

• $K^{n}$ (respectively, ${\widetilde {K}}^{n}$) 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 $\mathbb {Z} .$
• The spectrum of K-theory is $BU\times \mathbb {Z}$ (with the discrete topology on $\mathbb {Z}$), i.e. $K(X)\cong \left[X_{+},\mathbb {Z} \times BU\right],$ where [ , ] denotes pointed homotopy classes and BU is the colimit of the classifying spaces of the unitary groups: $BU(n)\cong \operatorname {Gr} \left(n,\mathbb {C} ^{\infty }\right).$ Similarly,
  $${\widetilde {K}}(X)\cong [X,\mathbb {Z} \times BU].$$
  For real K-theory use BO.
• There is a natural ring homomorphism $K^{0}(X)\to H^{2*}(X,\mathbb {Q} ),$ the Chern character, such that $K^{0}(X)\otimes \mathbb {Q} \to H^{2*}(X,\mathbb {Q} )$ 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
  $$K(X)\cong {\widetilde {K}}(T(E)),$$
  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:

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

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 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 $X$ with its rational cohomology. In particular, they showed that there exists a homomorphism

$ch:K_{\text{top}}^{*}(X)\otimes \mathbb {Q} \to H^{*}(X;\mathbb {Q} )$

such that

${\begin{aligned}K_{\text{top}}^{0}(X)\otimes \mathbb {Q} &\cong \bigoplus _{k}H^{2k}(X;\mathbb {Q} )\\K_{\text{top}}^{1}(X)\otimes \mathbb {Q} &\cong \bigoplus _{k}H^{2k+1}(X;\mathbb {Q} )\end{aligned}}$

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 (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. MR 1043170.
• 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. MR 2182598.
• 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".
• Stykow, Maxim (2013). "Connections of K-Theory to Geometry and Topology".