Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 141 / 1 / b / Solution

The solid torus supplied by the $\alpha$ side has fundamental group generated by the image $x$ of $\mu$, while $\lambda$ is trivial. The $\beta$ two-handle imposes $x^5=1$. The three-handles do not change the fundamental group, so