Homotopy invariance of vector bundles: If is compact Hausdorff and , then . Let be a homotopy between and , then ( is the projection map) and on are isomorphic. Let be such an isomorphism. Then is a section of on , so we can extend over all of . Note that in an open neighborhood of , we still get isomorphism. By compactness of , it includes , so the isomorphism type of is locally constant in , hence over all of . It remains true for paracompact base. The functor sending to rank -vector bundles on is representable by . The key point is that is contractible (Eilenberg-Mazur Swindle). The forms an algebra over the operad .
If we work over a compact base , then we just need to invert the trivial bundle to form since factors through some and we can take the orthogonal complement bundle of which satisfies . If we further mod out by the trivial bundle, then this reduced -group (vector bundlues modulo stable equivalence) is isomorphic to where . We can start build a cohomology theory and define . Note that we can rewrite . The upshot is that is contractible by computing its hommotopy group, since induces an isomorphism on for (by the fiber sequence), hence every homotopy group is zero when pass to the direct limit.
We can use Borel’s transgression theorem to compute the cohomology of (which is simply connected since its fundamental group is the component group of unitary group, which is trivial). Since is an exterior algebra on and they are transgressive by induction on , we get where is the transgression of and .
Bott periodicity: . For a short proof see here. See here for discussion of Bott periodicity. Since (see here), so and . Using the double periodicity, we can define and it constitutes a generalized cohomology theory.
Characteristic class: https://en.wikipedia.org/wiki/Characteristic_class (supposed to be a contravariant theory detecting existence of section)
Reference:
https://people.math.harvard.edu/~dafr/M392C-2012/Notes/lecture6.pdf