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I met a question asking me to classify the $2$-dimensional vector bundles of the sphere $S^2$.

I did not know how to classify the vector bundles in general. The only example I know was the line bundles of $S^1$: the cylinder and Moebius band. I guess this might be a result from the covering spaces, regarding how to glue the fibers on $\mathbb{R}$, but may not be true. Could anybody provide some inspiration using a concrete example?

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    Classifying the rank k vector bundles over the n-sphere is essentially the same approach for each appropriate k and n. For the case of rank two bundles on $S^2$ see for example [this SE answer](http://math.stackexchange.com/a/74918/12885)2012-09-05
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    More generally, for any nice space $X$, isomorphism classes of rank-$k$ vector bundles over $X$ are in bijection with homotopy classes of maps to $BO(k)=\mbox{colim }Gr_k(\mathbb{R}^N)$, the so-called *infinite Grassmannian* (which is also -- more suggestively -- called the *classifying space* for $O(k)$-bundles, i.e. bundles with structure group $O(k)$). This specializes to the clutching construction for spheres, and returns the general fact that $\pi_{n}(BG)\cong \pi_{n-1}(G)$.2012-09-06

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