TITLE: Vector bundles of rank $k$ with base $S_{1}$
QUESTION [8 upvotes]: I don't understand how to solve the following problem. Can you help me?
Prove that there are only two vector bundles of rank $k$ with base $S^{1}$ $-$ trivial $1_{k}$ and non-trivial $\eta_{k}$. 
Then let $S^{1}=\{z\in C, |z|=1\}$ and $f:S^{1} \to S^{1}$, $f(z)=z^{n}$. Find the bundle $f^{*}\eta_{k}$.

REPLY [2 votes]: You can also use the fact that $S^1$ is the 1-pt compactification of the unit interval
$I=[0,1]$, together with the fact that $I$ is contractible, so that every bundle over $I$ is trivial. Now, if you glue $0$~$1$ in $I$ to get $S^1$ from I , then the $\mathbb R^k$- bundle over $S^1$ will also glue along this map; informally, the fiber over $0$ will glue with the fiber over $1$ according to this map (which is called the monodromy). Like Jason said above, homotopic gluing maps give rise to isomorphic bundles. We then only need to consider the homotopy classes of homeomorphisms from I to itself , up to isotopy( so that we can glue the copy of $\mathbb R^k$ over $1$ with the copy of $\mathbb R^k$ over $0$. There are , up to isotopy, just two self-homeomorphisms; $+I$ and $-I$