It is known that,for a base space M,the equivalent classes of vector bundles over M generate a ring denoted by K0,with direct sum as + and tensor product as ×.
And we can give K0 a degree:all rank-n vector bundles are of degree n.
I am interested that:
for a topological manifold M, whether its Grothendieck ring K0 is generated by finite degrees or just finitely generated(true for spheres at least I guess)?
If not,what is a counter example?And when is it true?