Suppose $ξ$ and $η$ are two vector bundles over $X$. We say $ξ$ and $η$ are stably isomorphic if $ξ\oplus ε^{j}$ $\cong η\oplus ε^{k}$ for some j, k. Let $KX$ denote the set of stable isomorphism classes of bundles over $X$. [$X$ denotes either a manifold or a finite-dimensional simplicial or $CW$ complex; $ε^{i}$ denotes the trivial k-plane bundle.]
My question is how to prove "the operation of whitney sum induces a natural abelian group structure on $KX$ ". I couldn't prove this.
We have this theorem:
Let $ξ$ be a $C^{r}$ k-plane over an n-manifold $M$, $0 ≤ r ≤ ∞$.Then there is a $C^{r}$ n-plane bundle $η$ over $M$
such that $ξ\oplus η\cong_{r} M×R^{s}$.
This is a problem in the " differential topology of Morris W.Hirsch " (chapter4.classification).
Thanks for the help.