Topological K-theory explained

In mathematics, topological -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 -theory is due to Michael Atiyah and Friedrich Hirzebruch.

Definitions

Let 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 -vector bundles over under Whitney sum. Tensor product of bundles gives -theory a commutative ring structure. Without subscripts, K(X) usually denotes complex -theory whereas real -theory is sometimes written as KO(X) . The remaining discussion is focused on complex -theory.

As a first example, note that the -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 -theory,

\widetilde{K}(X) , defined for a compact pointed space (cf. reduced homology). This reduced theory is intuitively modulo trivial bundles. It is defined as the group of stable equivalence classes of bundles. Two bundles and are said to be stably isomorphic if there are trivial bundles \varepsilon _{1} and \varepsilon _{2} , so that E ⊕ \varepsilon _{1} \cong F ⊕ \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 into .

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

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

extends to a long exact sequence

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

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

\widetilde{K} ^{-n} (X): = \widetilde{K}(S ^{nX),} 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 -theory over contractible spaces is always \Z.

• The spectrum of -theory is

BU x \Z (with the discrete topology on \Z ), i.e. K(X) \cong \left [ X _{+,} \Z x BU \right ], where denotes pointed homotopy classes and is the colimit of the classifying spaces of the unitary groups: BU(n) \cong \operatorname{Gr} \left (n, \Complex ^{infty} \right ). Similarly, $$\backslash widetilde(X)\; \backslash cong\; [X,\; \backslash Z\; \backslash times\; BU].$$ For real -theory use .K ^{0(X)} \to H ^{2*} (X, \Q), the Chern character, such that K ^{0(X)} ⊗ \Q \to H ^{2*} (X, \Q) is an isomorphism.

• The equivalent of the Steenrod operations in -theory are the Adams operations. They can be used to define characteristic classes in topological -theory.
• The Splitting principle of topological -theory allows one to reduce statements about arbitrary vector bundles to statements about sums of line bundles.
• The Thom isomorphism theorem in topological -theory is $$K(X)\backslash cong\backslash widetilde(T(E)),$$ where is the Thom space of the vector bundle over . This holds whenever is a spin-bundle.
• The Atiyah-Hirzebruch spectral sequence allows computation of -groups from ordinary cohomology groups.
• Topological -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 x S ^{2)} = K(X) ⊗ K( S ^{2),} and K( S ^{2)} = \Z[H]/(H -1 ) ^{2} where H is the class of the tautological bundle on S ^{2} = P ^{1(\Complex),} i.e. the Riemann sphere.\widetilde{K} ^{n+2} (X) = \widetilde{K} ^{n(X).} \Omega ^{2} BU \cong BU x \Z.

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

Applications

Topological -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 homomorphismch : K ^{*} _{top} (X) ⊗ \Q \to H ^{*(X;\Q)}

such that

\begin{align} K ^{0} _{top} (X) ⊗ \Q & \cong o plus _{k} H ^{2k} (X;\Q) \\ K ^{1} _{top} (X) ⊗ \Q & \cong o plus _{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

• Atiyah–Hirzebruch spectral sequence (computational tool for finding K-theory groups)
• KR-theory
• Atiyah–Singer index theorem
• Snaith's theorem
• Algebraic K-theory

References

• Book: Atiyah . Michael Francis . Michael Atiyah . 1989 . K-theory . Advanced Book Classics . 2nd . . 978-0-201-09394-0 . 1043170.
• Book: Friedlander . Eric . Grayson . Daniel . 2005 . Handbook of K-Theory . Berlin, New York . . 978-3-540-30436-4 . 2182598 . 10.1007/978-3-540-27855-9.
• Book: Karoubi . Max . Max Karoubi . 1978 . K-theory: an introduction . Classics in Mathematics . Springer-Verlag . 0-387-08090-2 . 10.1007/978-3-540-79890-3.
• Karoubi . Max . Max Karoubi . 2006 . K-theory. An elementary introduction . math/0602082.
• Web site: Hatcher . Allen . Allen Hatcher . 2003 . Vector Bundles & K-Theory .
• Web site: Stykow . Maxim . Maxim Stykow . 2013 . Connections of K-Theory to Geometry and Topology .

Notes and References

1. Book: Hatcher. Vector Bundles and K-theory. 57. 27 July 2017.
2. Book: Adams, John . On the non-existence of elements of Hopf invariant one . 1960 . Ann. Math. 72 1.
3. Adams . John . 1962 . Vector Fields on Spheres . Annals of Mathematics . 75 . 3 . 603-632.