What sequence should I start with to calculate the cohomology of $\mathbb RP^n$?

Question

I want the simplest way to learn how to calculate the cohomology of $\mathbb RP^n$ with $\mathbb Z$ and $\mathbb Z_2$ coefficients, please?

What sequence should I start with?

Answer 1

In my opinion, the simplest way is to use the CW decomposition of $\mathbb{R}P^n$ with one cell in each dimension, resulting in the cellular chain complex as in Example 2.42 in Hatcher: $$ \dots \xrightarrow{0} \mathbb{Z} \xrightarrow{2} \mathbb{Z} \xrightarrow{0} \mathbb{Z} \xrightarrow{2} \mathbb{Z} \xrightarrow{0} \mathbb{Z} \to 0 $$ The right hand end of this is dimension $0$. The left hand end is unspecified because it depends on whether $n$ is even or odd: the first map is either $\times 0$ or $\times 2$ depending on the parity of $n$.

Hatcher explains the maps being multiplication by 0 or 2. Briefly and informally, you can think of this cell structure as coming from the cell structure on $S^n$ with two cells in each dimension, an upper and lower hemisphere in each dimension $k \leq n$. The map $S^n \to \mathbb{R}P^n$ identifies these pairs of hemispheres. You can compute the maps on the cellular chain complex for $S^n$ using the geometry of $S^n$, and then apply the quotient map to get the chain complex for $\mathbb{R}P^n$.

Apply $\mathrm{Hom}(-, R)$ for any coefficient ring $R$ to get the corresponding cochain complex, from which it is easy to compute $H^*(\mathbb{R}P^n; R)$.