Let $G_n = \langle\tau\rangle$ and $G_m = \langle\sigma\rangle$ be groups of $n$-th and $m$-th roots of unity. Define action of $G_n \times G_m$ on $\mathbb{C}P^1$ as follows. $$ (\tau^k, \sigma^t) \cdot[x_0, x_1] = [\tau^kx_0, \sigma^tx_1]$$
I know that the only group that can act freely on sphere is $\mathbb{Z}/(2)$ so I almost certainly won't get $\mathbb{R}P^2$. People say something about branched coverings and Riemann surfaces claiming there always will be $S^2$ but I don't have this machinery. What I do have is the first half of Hatcher's algebraic topology technical tools. What can I do to prove it's a sphere?