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By introducing the "clutching function" one can relate the complex (real) vector bundles on a sphere with homotopy groups of $GL_n(\Bbb C)$ ($GL_n^+(\Bbb R)$ for oriented ones). Can we do a similar thing to torus? The motivation is that I'm curious about how many vector bundles of rank 1,2,3,... are there over our familiar spaces, at least for some specified low rank. (BTW, It seems in the case of 2, plane bundles correspond to $H^2(X,\Bbb Z)$?)

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    (An idea I failed to implement correctly: use Künneth theorem for K-theory.)2011-11-14
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    BTW, can someone prove that the answer is the same as for $S^1\vee S^1\vee S^2$? (The latter space is stably homotopy equivalent to the torus, so at least in the stable range everything follows from Bott periodicity etc. But surely there should be more elementary proof?)2011-11-14
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    @Grigory: It follows from my second answer that it's the same as for $S^1\vee S^1\vee S^2$, but it's a proof by directly computing both, so not very enlightening.2011-11-14

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