Stable cancellation for complex vector bundles

Let $E_0,E_1$ be complex vector bundles of rank $d\ge 1$ over a finite CW complex $X$ with $\dim X\le 2d$. Iffor some $r$, then $E_0\cong E_1$: in this dimension range, stable isomorphism already determines the actual vector bundle.

The stabilization $BU(d)\to BU$ of the classifying space for rank-$d$ complex vector bundles is an n-connected map with $n=2d+1$. One sees the range from the fibration with fiber $U(d+1)/U(d)\cong S^{2d+1}$ and then iterates stabilization. The vanishing of the fiber's homotopy groups through degree $2d$ allows a stable homotopy between classifying maps to lift over $X\times [0,1]$, relative to its ends. Thus the original classifying maps are homotopic and the bundles are isomorphic.

The rank bound is useful when invariants such as the Chern character first identify stable classes: it supplies the final cancellation step in the stated rank and dimension range.

Two rank-$d$ complex vector bundles over ${\mathbb{CP}}^d$ are isomorphic exactly when their Chern classes agree. Equality of Chern classes gives equality of their Chern characters; the Chern character is injective here because $K^0({\mathbb{CP}}^d)$ is torsion-free. The bundles are therefore stably isomorphic, and stable cancellation for complex vector bundles applies because ${\mathbb{CP}}^d$ has real dimension $2d$.