punctured Mobius band in high dimension

Question

Let $M^{n+1}$ be the non-trivial line bundle over sphere $S^n$. When $n=1$, $M^2$ is Mobius band and the punctured space $M^2\setminus *\simeq S^1$. How about $M^{n+1}\setminus *$ in general? Does $M^{n+1}\setminus *\simeq S^n$?

Answer 1

Every line bundle is trivial if it is orientable. But if the first cohomology of the space is trivial (which is true for simply connected spaces), the bundle will be orientable, since the first Stiefel Whitney class is an obstruction to this. Hence any line bundle over a higher dimensional sphere is trivial.