User knot - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T21:46:46Z http://mathoverflow.net/feeds/user/1057 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/2671/how-are-vector-bundles-and-homotopy-groups-related (how) are vector bundles and homotopy groups related? knot 2009-10-26T20:18:25Z 2009-10-26T21:11:36Z <p>Hello,</p> <p>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}?</p> http://mathoverflow.net/questions/2340/what-is-the-first-interesting-theorem-in-insert-subject-here/2456#2456 Answer by knot for What is the first interesting theorem in (insert subject here)? knot 2009-10-25T11:11:30Z 2009-10-25T11:11:30Z <p>Homotopy Theory: the Hopf Fibration?</p> http://mathoverflow.net/questions/2671/how-are-vector-bundles-and-homotopy-groups-related/2674#2674 Comment by knot knot 2009-10-26T20:55:19Z 2009-10-26T20:55:19Z Thank you (both) very much, now it makes sense!