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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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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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    (Stably parallelizable = tangent bundle + some trivial bundle is trivial)2012-02-10