For any $N \in \mathbb N_1$, The map $SO(\mathbb R,2) \to SO(\mathbb R,2) \ R \mapsto R^N$ is an N fold covering map. This is easy for me to see by considering the rotations as complex numbers of magnitude one. I would like to know if this result generalises to higher dimensions. That is:
Is for all n > 1 the map $SO(\mathbb R,n) \to SO(\mathbb R,n) \ R \mapsto R^N$ an N-fold covering?
If it is the case I'm not really sure how to prove it as I don't have an analog for the complex numbers in higher dimensions. Indeed looking at Wikipedia to find representations of $SO(R,n)$ does not really lead to a discernible pattern for what $SO(R,n)$ may be for differing $n$. Thus I am a stuck with how to proceed.
Owing to comment made by Lord Shark the Unknown I will fill in the details for an anwser:
For $n \geq 3$ the hypothesis fails.
Conider the case $N = 4$. there are unaccountably infinite rotations which when raised to the fourth power leave the identity. To observe this, take any unit vector $v$ that is orthogonal to $e_1$. Consider the unique rotation $R$ in the $<x,v>$ plane taking $x$ to $v$. It is such that $R^4 = I$. As there are uncountable many such $v$ all leading to different $R$ we have found uncountable many rotations that have fourth power the identity and the result is proved.
Thanks for the help Lord Shark!