Topological K-theory

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.

## Definitions

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Let X be a compact Hausdorff space and  k = R {\displaystyle k=\mathbb {R} } or  C {\displaystyle \mathbb {C} } . Then  K k ( X ) {\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 ) {\displaystyle K(X)} usually denotes complex K-theory whereas real K-theory is sometimes written as  K O ( X ) {\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,  K ~ ( X ) {\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  ε 1 {\displaystyle \varepsilon _{1}} and  ε 2 {\displaystyle \varepsilon _{2}} , so that  E ⊕ ε 1 ≅ F ⊕ ε 2 {\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,  K ~ ( X ) {\displaystyle {\widetilde {K}}(X)} can be defined as the kernel of the map  K ( X ) -> K ( x 0 ) ≅ Z {\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)

K ~ ( X / A ) -> K ~ ( X ) -> K ~ ( A ) {\displaystyle {\widetilde {K}}(X/A)\to {\widetilde {K}}(X)\to {\widetilde {K}}(A)}

extends to a long exact sequence

⋯ -> K ~ ( S X ) -> K ~ ( S A ) -> K ~ ( X / A ) -> K ~ ( X ) -> K ~ ( A ) . {\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

K ~ − n ( X ) := K ~ ( S n X ) , n ≥ 0. {\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:

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

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

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

## Properties

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* K n {\displaystyle K^{n}} (respectively,  K ~ n {\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  Z . {\displaystyle \mathbb {Z} .}
* The spectrum of K-theory is  B U × Z {\displaystyle BU\times \mathbb {Z} } (with the discrete topology on  Z {\displaystyle \mathbb {Z} } ), i.e.  K ( X ) ≅ [ X + , Z × B U ] , {\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:  B U ( n ) ≅ Gr ⁡ ( n , C ∞ ) . {\displaystyle BU(n)\cong \operatorname {Gr} \left(n,\mathbb {C} ^{\infty }\right).} Similarly,  K ~ ( X ) ≅ [ X , Z × B U ] . {\displaystyle {\widetilde {K}}(X)\cong [X,\mathbb {Z} \times BU].} For real K-theory use BO.
* There is a natural ring homomorphism K 0 ( X ) -> H 2 ∗ ( X , Q ) , {\displaystyle K^{0}(X)\to H^{2*}(X,\mathbb {Q} ),} the Chern character, such that  K 0 ( X ) ⊗ Q -> H 2 ∗ ( X , Q ) {\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 ) ≅ K ~ ( T ( E ) ) , {\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

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The phenomenon of periodicity named after Raoul Bott (see Bott periodicity theorem) can be formulated this way:

* K ( X × S 2 ) = K ( X ) ⊗ K ( S 2 ) , {\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}} where H is the class of the tautological bundle on  S 2 = P 1 ( C ) , {\displaystyle \mathbb {S} ^{2}=\mathbb {P} ^{1}(\mathbb {C} ),} i.e. the Riemann sphere.
* K ~ n + 2 ( X ) = K ~ n ( X ) . {\displaystyle {\widetilde {K}}^{n+2}(X)={\widetilde {K}}^{n}(X).}
* Ω 2 B U ≅ B U × Z . {\displaystyle \Omega ^{2}BU\cong BU\times \mathbb {Z} .}

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

## Applications

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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

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Michael Atiyah and Friedrich Hirzebruch proved a theorem relating the topological K-theory of a finite CW complex  X {\displaystyle X} with its rational cohomology. In particular, they showed that there exists a homomorphism

c h : K top ∗ ( X ) ⊗ Q -> H ∗ ( X ; Q ) {\displaystyle ch:K_{\text{top}}^{*}(X)\otimes \mathbb {Q} \to H^{*}(X;\mathbb {Q} )}

such that

K top 0 ( X ) ⊗ Q ≅ ⨁ k H 2 k ( X ; Q ) K top 1 ( X ) ⊗ Q ≅ ⨁ k H 2 k + 1 ( X ; Q ) {\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  X {\displaystyle X} .

## See also

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

## References

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1. ^ Hatcher. Vector Bundles and K-theory (PDF). 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.

* Atiyah, Michael Francis (1989). K-theory. Advanced Book Classics (2nd ed.). Addison-Wesley. ISBN 978-0-201-09394-0. MR 1043170.
* 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. MR 2182598.
* 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".
* Stykow, Maxim (2013). "Connections of K-Theory to Geometry and Topology".