= Stable cancellation for complex vector bundles
Let $E_0,E_1$ be <complex vector bundles> of rank $d\geq1$ over a <finite CW complex> $X$ with $\dim X\leq2d$. If
$$
E_0\oplus\mathbb C^r\cong E_1\oplus\mathbb C^r
$$
for 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.
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