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= \R or \Complex. 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 \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^nX), \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, \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.
  • The spectrum of K-theory is BU\times\Z (with the discrete topology on \Z), i.e. K(X) \cong [ X_+, \Z \times BU ], where [ , ] denotes pointed homotopy classes and BU is the colimit of the classifying spaces of the unitary groups: BU(n) \cong \operatorname{Gr} (n, \Complex^{\infty} ). Similarly,
\widetilde{K}(X) \cong [X, \Z \times BU].

For real K-theory use BO.

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.

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

In real K-theory there is a similar periodicity, but modulo 8.

Applications

Topological K-theory has been applied in John Frank Adams’ proof of the “Hopf invariant one” problem via Adams operations.[2] Adams also proved an upper bound for the number of linearly-independent vector fields on spheres.[3]

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\Q \to H^*(X;\Q)

such that

\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}

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

See also

References

  1. ^ Hatcher. Vector Bundles and K-theory. p. 57. Retrieved 27 July 2017.
  2. ^ Adams, John (1960). On the non-existence of elements of Hopf invariant one. Ann. Math. 72 1.
  3. ^ Adams, John (1962). "Vector Fields on Spheres". Annals of Mathematics. 75 (3): 603–632. doi:10.2307/1970213. JSTOR 1970213