I am trying to read Hatcher's vector bundle and $K$-theory. I am trying to understand the obstruction theory in Chapter $3$.
When we get a vector bundle $\pi:E\rightarrow B$ and we may suppose that $B$ is $CW$ complex. We first suppose that $E$ is trivial on $B^1$, and we are tring to extend the $n$-sections over $B^2$. In Hatcher's original remark, we can get a map $\partial D^2\rightarrow SO(n)$ via characteristic map. (This is done by pulling bundle to $D^2$. this gets a trival bundle and we can restrict on the boundary $\partial D^2$, I think).
But here is the remark I cound't understand totally. This procedure defines an element $\pi_1(SO(n))$. By the definition of fundamental group, this expresses an equivalence classes of loops up to homotopy in $SO(n)$, but where does the homotopy come from ? (More generally, for the group $\pi_{n-k}(V_k(\mathbb R^n))$)
I think our goal is to extend the sections over $B^2$, just imagine a torus $T^2$, we've find a section over two circles ($1$-cell), and when we try to attach $2$-cell(a long "tube" with many circles I think) on the, when we find a new sections on the boundary, we can turn this around by $360$°. This procedure describes what hatcher means $\partial D^2\rightarrow SO(n)$. ( This is just the special case of $V_{n-1}(\mathbb R^n)\cong SO(n)$ I guess).
But I still can't see why this defines a fundamental group ? Where does the homotopy comes from ?
Hope I've made my script clear. Appeciate your feedback!