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

The standard CW complex structure on Real projective space ${\mathbb{RP}}^n$ has one cell in each dimension $0,1,\dots ,n$. Its cellular chain complex has $C_q\cong \mathbb{Z}$ and differentialConsequentlyAfter tensoring the cellular complex with ${\mathbb{F}}_2=\mathbb{Z}/2$, every differential vanishes, so

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

The quotient is the mapping torus ofA mapping torus of a cellular map has a finite CW structure. On integral homology, ${\tau }_{\ast }$ is $1$ on $H_0(T^2)$ and $H_2(T^2)$ and is $-I$ on $H_1(T^2)\cong {\mathbb{Z}}^2$. The Wang sequence therefore givesOver ${\mathbb{F}}_2$, the map ${\tau }_{\ast }$ is the identity. The Wang sequence splits as vector spaces into one copy of $H_q(T^2;{\mathbb{F}}_2)$ and one copy of $H_{q-1}(T^2;{\mathbb{F}}_2)$, yielding