A sphere covering a space with fundamental group Z/5Z

Question

I am to find a space whose universal cover is a sphere and its fundamental group is Z/5Z. Does anybody have an idea how to approach the problem?

Answer 1

The fundamental group of the lens space $L^{2n+1}(m)$ is given by the cyclic group of order $m$, i.e., $$ \pi_1(L^{2n+1}(m)) = C_m. $$ Consider the covering map $p\colon S^{2n+1}\rightarrow L^{2n+1}(m)$. Now take $m=5$. For details see, for example, here. The action of $C_m$ on $S^{2n+1}$ is clearly free, so the quotient map is a covering map with deck group $C_m$. Since $S^{2n+1}$ is even simply connected, it is the universal cover of this lens space.