Definition of characteristic element of a vector bundle? Is this the same as the characteristic class?

Question

I'm reading Kosinski's book "Differential Manifolds," and he uses a term which he never defines. He refers repeatedly to the "characteristic element" of a k-dimensional vector bundle, which is apparently an element of $\pi_{k-1}(SO(k))$. When I try to track down a definition, I only turn up information about the characteristic classes of vector bundles, but these are elements of the cohomology groups of the base. Is anyone familiar with this use of the term "characteristic element?"

Answer 1

Given a bundle (let's say vector bundle for now) over $\Sigma X$, $X$ a pointed space, pick a trivialization over the two cones: $C_+X$ and $C_-X$ (We demand for convenience these trivializations agree at the basepoint, but that doesn't matter much). These trivializations, at a point $x \in X = C_+X \cap C_-X$ differ by an element of the structure group of the vector bundle, $O(k)$. Assembling all these together, we get a pointed map $X \to O(k)$ called a 'clutching function'. Now you can verify that the homotopy class of this map doesn't depend on our choice of trivialization, and that homotopic clutching functions come from/give rise to isomorphic vector bundles. So there is a bijection between vector bundles over $\Sigma X$ and homotopy classes $[X,O(k)]$.

Now take $X = S^{n-1}$ to get your case (usually the most important case); Kosinki is allowed to say $SO(k)$ because, when $n>1$, $S^{n-1}$ is connected and must have image lying in the identity component of $O(k)$.