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 ${\displaystyle \mathbb {C} }$ . 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, ${\displaystyle K(X)}$  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 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, ${\displaystyle {\widetilde {K}}(X)}$ , 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 ${\displaystyle \varepsilon _{1}}$  and ${\displaystyle \varepsilon _{2}}$ , 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, ${\displaystyle {\widetilde {K}}(X)}$  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)

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

extends to a long exact sequence

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

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

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

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:

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

Here ${\displaystyle X_{+}}$  is ${\displaystyle X}$  with a disjoint basepoint labeled '+' adjoined.[1]

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

Properties

• ${\displaystyle K^{n}}$  (respectively, ${\displaystyle {\widetilde {K}}^{n}}$ ) 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 ${\displaystyle \mathbb {Z} .}$
• The spectrum of K-theory is ${\displaystyle BU\times \mathbb {Z} }$  (with the discrete topology on ${\displaystyle \mathbb {Z} }$ ), 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
${\displaystyle K(X)\cong {\widetilde {K}}(T(E)),}$

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

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 ${\displaystyle X}$ .