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Topological K-theory - Wikipedia

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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.

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.

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

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

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]

Michael Atiyah and Friedrich Hirzebruch proved a theorem relating the topological K-theory of a finite 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}.

  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.