Today I cam across the following exercise:
exercise
Let $\alpha: \mathbb R^n \rightarrow \mathbb R^n$ be a map with $\alpha^p=1$ for some prime $p$. Show that $\alpha$ has a fixed point.
origin
I found the exercise somewhere in my notes. The setting was roughly orientation and the orientation bundle.
approach
For my approach, of course I assumed that $\alpha$ has no fixed point and just as in the Brouwer fixed point theorem I obtained a map $\mathbb R^n \rightarrow \mathbb S^{n-1}$. But now I don't know how to continue. I especially have trouble using the assumption that $\alpha^p=1$.
Any hint or help for this exercise is welcome!
Sincerely Slin
Edit
This indeed is a duplicate of this
The answer uses some more advanced theory. I am interested how to do it the way it is suggested in Bredon's "Topology and Geometry". I cannot make the connection between $\mathbb R^n$ and coverings.