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 \( {\displaystyle k=\mathbb {R} } \) or \( \mathbb{C} \) . Then \( {\displaystyle K_{k}(X)} \) 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 \( {\displaystyle KO(X)} \). The remaining discussion is focused on complex K-theory.

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 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 \( {\displaystyle E\oplus \varepsilon _{1}\cong F\oplus \varepsilon _{2}} \). 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 \( {\displaystyle K(X)\to K(x_{0})\cong \mathbb {Z} } \) 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.[1]

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

\( K^{n} \) (respectively, K ~ n {\displaystyle {\widetilde {K}}^{n}} \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 Z . {\displaystyle \mathbb {Z} .} {\displaystyle \mathbb {Z} .} \)

The spectrum of K-theory is \( {\displaystyle BU\times \mathbb {Z} } \) (with the discrete topology on \( \mathbb {Z} ) \) , i.e. \( {\displaystyle K(X)\cong \left[X^{+},\mathbb {Z} \times BU\right],} \) where [ , ] denotes pointed homotopy classes and BU is the colimit of the classifying spaces of the unitary groups: \( {\displaystyle BU(n)\cong \operatorname {Gr} \left(n,\mathbb {C} ^{\infty }\right).} \) Similarly,

\( {\displaystyle {\widetilde {K}}(X)\cong [X,\mathbb {Z} \times BU].} \)

For real K-theory use BO.

There is a natural ring homomorphism \( {\displaystyle K^{0}(X)\to H^{2*}(X,\mathbb {Q} ),} \) the Chern character, such that \( {\displaystyle K^{0}(X)\otimes \mathbb {Q} \to H^{2*}(X,\mathbb {Q} )} \) 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:

\( {\displaystyle K(X\times \mathbb {S} ^{2})=K(X)\otimes K(\mathbb {S} ^{2}),} \) and K ( S 2 ) = Z [ H ] / ( H − 1 ) 2 {\displaystyle K(\mathbb {S} ^{2})=\mathbb {Z} [H]/(H-1)^{2}} {\displaystyle K(\mathbb {S} ^{2})=\mathbb {Z} [H]/(H-1)^{2}} where H is the class of the tautological bundle on S 2 = P 1 ( C ) , {\displaystyle \mathbb {S} ^{2}=\mathbb {P} ^{1}(\mathbb {C} ),} {\displaystyle \mathbb {S} ^{2}=\mathbb {P} ^{1}(\mathbb {C} ),} i.e. the Riemann sphere.

\( \widetilde {K}^{{n+2}}(X)=\widetilde {K}^{n}(X). \)

\( {\displaystyle \Omega ^{2}BU\cong BU\times \mathbb {Z} .} \)

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 CW complex {\displaystyle X} with its rational cohomology. In particular, they showed that there exists a homomorphism

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

such that

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

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