When is a virtual bundle an actual bundle?

As far as I understand, one can make a monoid from the space of vector bundles on a (compact) manifold $$M$$, with respect to the direct sum operation $$\oplus$$. In order to make this into a group, one needs to add 'inverses' and this leads to the consideration of formal differences $$V-W$$ of vector bundles. One can again compute Euler classes, Chern classes, etc... for these objects called 'virtual bundles'.

Now my question is:

When is such an object an actual vector bundle? Is there an easy criterion to check? For instance, if I have a complex virtual vector bundle and I know that its Chern classes are integral i.e. in $$H^*(M,\mathbb{Z})$$, rather than $$H^*(M,\mathbb{Q})$$, is that sufficient?

Disclaimer: I am only familiar with basic aspects of $$K$$-theory, so the above question might be completely trivial, but I couldn't find an answer.

• Let $c$ denote the total Chern class. Then the Whitney formula is $c(E\oplus F)=c(E)c(F)$. So, to be consistent, I think you should have $c(E-F)=c(E)c(F)^{-1}=c(E)\cdot 1/(1+c_1(F)+c_2(F)+\ldots + c_r(F))=c(E)\sum_{k\geq 0}(-1)^k(\sum_{i=1}^r c_i(F))^k$ which seems integral. Feb 9 '20 at 14:37
• @Qfwfq 's argument shows that it is necessary to have $c_i(E)=0$ for $i$ large enough. But it is clearly not sufficient because any bundle of the form $V-nC$, where $nC$ is the trivial $n$-dimensional complex bundle verifies this. Feb 9 '20 at 15:42
• Over a compact space every map $f: X \to BO$ factors through a finite-dimensional Grassmannian. In particular every virtual v.b. is of the form $V - n$ for some vector bundle $V$, and $n$ is determined by the rank of the virtual vb.
– mme
Feb 9 '20 at 16:19
• And once you've used Mike Miller's observation, a necessary condition is that the last 𝑛 chern classes of 𝑉 are zero. If the rank and the dimension of the manifold interact well you can appeal to some obstruction theory, but I think it's a pretty hard problem in general, since it's the same as asking for $n$ linearly independent sections of $V$. Feb 9 '20 at 21:23