Let $M$ be an orientable $3$-manifold with connected boundary $\Sigma_g$, a surface of genus $g>0$.
I would like to find a reference to the following two statements.
1) $\pi_1(M)\ne 0$.
2) $\pi_1(M)\ne \pi_1(\Sigma_g)$.
2)' If 2) is too hard I would be happy just to know that the map $\pi_1(\Sigma_g)\to M$ is not an isomorphism.
I think I can prove the first statement by contradiction. If $\pi_1(M)$ were trivial it would stay so after gluing a handlebody to $\Sigma_g$. In the resulting simply-connected manifold one can chose loops inside $M$ that have non-zero linking number with loops inside the handlebody hence they are not null-homologous in $M$. Hence we get a contradiction with $\pi_1(M)=0$.
But I don't see how to prove 2), and this might be hard. I would be grateful for some tips.