Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 114 / 1 / ii / Solution

The involution $(z_1,z_2)\mapsto (-z_1,{\overline{z}}_2)$ is free. Taking the first circle modulo the half-turn exhibits the quotient as the mapping torus of a reflection of $S^1$, hence as the Klein bottle. It has a finite CW structure with one zero-cell, two one-cells, and one two-cell. With suitable generators its integral cellular differential isThereforewhereas reduction modulo two kills the only nonzero boundary and gives