Hatcher-Algebraic-Topology · Problem 1.3.21

The key is using the Seifert–van Kampen theorem to compute the fundamental group by decomposing each space into simpler pieces whose groups and intersection you understand. For $X$, the torus contributes two commuting generators while the Möbius band (which is homotopy-equivalent to $S^1$) adds a third generator, but the twist in the Möbius band forces a relation between how this new generator interacts with the torus generators. For $Y$, $\mathbb{RP}^2$ already has $\pi_1 = \mathbb{Z}/2\mathbb{Z}$, and attaching the Möbius band—which also has a twist—creates a new relation. Once you have the presentation, the universal cover is the covering space corresponding to the kernel of the abelianization map, and the deck transformations are determined by how $\pi_1$ acts on the lifts of your generators.

The key is using the Seifert–van Kampen theorem to compute the fundamental group by decomposing each space into simpler pieces whose groups and intersection you understand. For $X$, the torus contributes two commuting generators while the Möbius band (which is homotopy-equivalent to $S^1$) adds a third generator, but the twist in the Möbius band forces a relation between how this new generator interacts with the torus generators. For $Y$, $\mathbb{RP}^2$ already has $\pi_1 = \mathbb{Z}/2\mathbb{Z}$, and attaching the Möbius band—which also has a twist—creates a new relation. Once you have the presentation, the universal cover is the covering space corresponding to the kernel of the abelianization map, and the deck transformations are determined by how $\pi_1$ acts on the lifts of your generators.