nLab
virtual vector bundle

Contents

Idea

A formal difference (with respect to direct sum/Whitney sum) of vector bundles. Appears as representative for classes in topological K-theory. A virtual vector bundle [E][F][E] - [F] has a virtual rank rk([E][F])rk([E]-[F]), which is the integer rk(E)rk(F)rk(E) - rk(F).

Properties

Proposition

Let X be a space that is locally compact and regular. Then any virtual vector bundle of virtual rank 0 has some tensor power that is 0 in K-theory.

The following argument comes from (Karoubi 1978, Theorem II.5.9), but generalised to take into account non-compact spaces which are introduced in the following pages.

Proof

For closed subsets Y 1Y_1 and Y 2Y_2, then the ring structure on K-theory means

K Y 1(X)K Y 2(X)K Y 1Y 2(X) K_{Y_1}(X)\cdot K_{Y_2}(X) \subset K_{Y_1\cup Y_2}(X)

where K Y i(X):=ker(K(X)K(Y i))K_{Y_i}(X) := ker(K(X) \to K(Y_i)). We can likewise consider the same for a sequence Y 1,,Y nY_1,\ldots ,Y_n.

If e=[E][rk(E)̲]e = [E] - [\underline{rk(E)}] is in ker(rk)ker(rk) (i.e. reduced K-theory K˜(X)\tilde K(X)) with supp(e)supp(e) in some compact set CC (recall that vector bundle K-theory elements are differences of vector bundles that are isomorphic outside a compact set. Without loss of generality such an element can be represented as [E][V̲][E] - [\underline{V}] where V̲\underline{V} is a trivial bundle with fibre VV), cover CC by finitely many closed neighbourhoods Y 1,,Y nY_1,\ldots ,Y_n over which EE trivialises. Thus ee is in K Y i(X)K_{Y_i}(X) for each i=1,ni=1,\ldots n.

Thus ee is in the intersection of all the K Y i(X)K_{Y_i}(X) as well as K Z(X)K_Z(X) for ZZ the complement of supp(e)supp(e). Thus e n+1e^{n+1} is in K Y iZ(X)=K X(X)=0K_{\cup Y_i \cup Z}(X) = K_X(X) = 0.

Hence for every element in ker(rk)ker(rk), some power of it is zero, hence ker(rk)ker(rk) is a nil ideal.

References

  • Max Karoubi, K-Theory: An Introduction, Grundlehren der mathematischen Wissenschaften 226, (1978) Springer-Verlag

Revised on August 18, 2016 04:40:58 by David Roberts (58.179.228.38)