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Let $X$ be a paracompact and well behaved space. Topological K-Theory $K(X)$ of $X$ is group completing the monoid of isomorphism classes of vector bundles over $X$ with the Whitney sum.

Two vector bundles $E,E'\to X$ represent the same class $[E]=[E']$ in $K(X)$ if there is (after applying some theory) a trivial bundle $K$ such that $K\oplus E=K\oplus E'$.

What is an example of two non-isomorphic vector bundles $E$, $E'$ such that $[E]=[E']$?

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up vote 5 down vote accepted

The tangent vector bundle $\tau$ of $S^2$ gives such example: it's non-trivial (since there are no non-vanishing vector fields on $S^2$) but $\tau\oplus 1=3$ ($1$ is the normal bundle, which is trivial).

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More generally, all spheres are stable parallelizable in this manner. More exotically: any time you have a smooth map $S^{n+k} \rightarrow S^n$, the preimage of a regular value is stably parallelizable. – Dylan Wilson Feb 10 '12 at 16:06
(Stably parallelizable = tangent bundle + some trivial bundle is trivial) – Dylan Wilson Feb 10 '12 at 16:07

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