Covering $S_2$ with $S_3$(or $S_n$)

Question

How can I construct a covering map $p : S_3 \to S_2$? I can construct coverings for $S^1\vee S^1$, but same ideas don't work for $S_2$ ($S_g$ is sphere with $g$ handles).

Answer 1

Consider the $\Bbb Z_2$-action on $\Sigma_3$ by rotational symmetries, switching (by rotation, so that the two small circles switch places too) the copies of doubly punctured torii bounded by the small circles marked on the surface.

I claim that the quotient map $p : \Sigma_3 \to \Sigma_3/\Bbb Z_2 \cong \Sigma_2$ is the desired covering map (see pg 73 in Hatcher's book for a better picture of the analogous covering map $\Sigma_{11} \to \Sigma_3$)

Fact : If a discrete group $G$ acts properly discontinuously and freely on a topological space $X$, then the quotient map $\pi : X \to X/G$ is a covering map.

The last bit of the problem is to note that $\Sigma_3/\Bbb Z_2 \cong \Sigma_2$, which is obvious since taking one of the doubly punctured torii and pasting the two disks those little circles bound gives us the orbit space, which is indeed the double torus.