[Math] (how) are vector bundles and homotopy groups related

Question

Hello,

homotopic maps induce isomorphic pullback bundles, and so isomorphism classes of vector bundles over X correspond to homotopy classes of maps from the grassmannian to X. But I think, in the case of X = S^1 (the 1-sphere), the isomorphism classes of 1-bundles correspond to (the generators of) π___{1}(S^1), since there is the trivial bundle, the Moebius bundle, and that's it. So my question is: am I right with this, and if yes: what needs to happen that π_n(X) = {homotopy classes of maps:grassmannian -> X}?

Answer 1

The map goes the other way: vector bundles over X correspond to homotopy classes of maps from X into a grassmannian.

Let BO(n) be the grassmannian of n-plane bundles in R-infinity. Then, if you want to know about n-dimensional real vector bundles over S^k you are led to study the homotopy classes of maps from S^k to BO(n), or in other words π_k BO(n).

In particular, the fact that there are exactly two 1-bundles over S¹ comes from a calculation π₁ BO(1)=Z/2.