Topological K-theory

Short description: Branch of algebraic topology

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.

## Contents

* 1 Definitions
* 2 Properties
* 3 Bott periodicity
* 4 Applications
* 5 Chern character
* 6 See also
* 7 References

## Definitions

Let X be a compact Hausdorff space and k=ℝ or ℂ. Then Kk(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, 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 ε1 and ε2, so that E⊕ε1≅F⊕ε2. 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, K~(X) can be defined as the kernel of the map K(X)->K(x0)≅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)

K~(X/A)->K~(X)->K~(A)

extends to a long exact sequence

⋯->K~(SX)->K~(SA)->K~(X/A)->K~(X)->K~(A).

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

K~−n(X):=K~(SnX),n≥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)=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

* Kn (respectively, 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×ℤ (with the discrete topology on ℤ), i.e. K(X)≅[X+,ℤ×BU], where [ , ] denotes pointed homotopy classes and BU is the colimit of the classifying spaces of the unitary groups: BU(n)≅Gr⁡(n,ℂ∞). Similarly, K~(X)≅[X,ℤ×BU]. For real K-theory use BO.
* There is a natural ring homomorphism K0(X)->H2*(X,ℚ), the Chern character, such that K0(X)⊗ℚ->H2*(X,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)≅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×𝕊2)=K(X)⊗K(𝕊2), and K(𝕊2)=ℤ[H]/(H−1)2 where H is the class of the tautological bundle on 𝕊2=ℙ1(ℂ), i.e. the Riemann sphere.
* K~n+2(X)=K~n(X).
* Ω2BU≅BU×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:Ktop*(X)⊗ℚ->H*(X;ℚ)

such that

Ktop0(X)⊗ℚ≅⨁kH2k(X;ℚ)Ktop1(X)⊗ℚ≅⨁kH2k+1(X;ℚ)

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

1. ↑ Hatcher. Vector Bundles and K-theory. pp. 57. https://pi.math.cornell.edu/~hatcher/VBKT/VB.pdf. 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.

* Atiyah, Michael Francis (1989). K-theory. Advanced Book Classics (2nd ed.). Addison-Wesley. ISBN 978-0-201-09394-0.
* 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.
* 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". https://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html.
* Stykow, Maxim (2013). "Connections of K-Theory to Geometry and Topology". https://www.researchgate.net/publication/330505308.

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