# Classify the vector bundles of a manifold.

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 – Ben Sep 5 '12 at 16:10
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 $\pi_{n}(BG)\cong \pi_{n-1}(G)$. – Aaron Mazel-Gee Sep 6 '12 at 13:52

The idea for bundles over the spheres could be to say that you can cover the sphere by the two hemispheres, which are contractible. Hence any bundle restricted to these will be trivial and essentially the bundle is determined by how you glue the two trivial bundles together on the equator. Hence 2-dimensional bundles over $S^2$ correspond bijectively to $\pi_1(SO(2)) \cong \mathbb{Z}$.