Let $S_{g,n}^b$ be a genus $g$ surface with $b$ boundary components and $n$ punctures. I'm having some trouble with these past qualifying exam questions:
Is there a covering map $p\colon \Sigma_3^1\to \Sigma_2^1$?
Is there a covering map $p\colon \Sigma_{3,1}\to\Sigma_{2,1}?$
I believe that the answer is no in both cases. Both $\Sigma_{3,1}$ and $Sigma_3^1$ deformation retracts onto a wedge of $6$ circles, while $\Sigma_{2,1}$ and $\Sigma_2^1$ deformation retract onto a wedge of $4$ circles. Any covering $p\colon\Sigma_3^1\to\Sigma_2^1$ would then induce an injection $p_*\colon F_6\to F_4$ whose image has finite index, because the target is compact. I know that we can inject $F_6\hookrightarrow F_4$, but I don't think the image can have finite index. I don't know how to prove this, though.
I haven't made much progress on the first question. Any help would be greatly appreciated.