Solution
= Solution
The space $V$ is homeomorphic to $\mathbb R^3$, hence contractible and simply connected. Part (a) therefore identifies $V\to N$ as the <universal covering map>. Its <deck transformation group> consists of the left translations by $\Gamma$, so covering-space theory gives
$$
\pi_1(N)\cong\Gamma.
$$
Combining this with the homeomorphism from part (c),
$$
\boxed{\ \pi_1(M)\cong\Gamma.\ }
$$