Here is a question from Hatcher (2.2.19):
I assume that those $H_i$'s are homology groups, Hatcher denotes both the chain complexes and the homology groups by $H$ (I will denote the chain complexes by $C_i$).
$\dots \xrightarrow{d_{n+1}} C_n(\Bbb{R}P^n/\Bbb{R}P^m) \xrightarrow{d_n} C_{n-1}(\Bbb{R}P^{n-1}/\Bbb{R}P^m)\xrightarrow{d_{n-1}} \dots \xrightarrow{d_{m+1}} C_m(\Bbb{R}P^m/\Bbb{R}P^m) \xrightarrow{d_m} 0$
All of them are isomorphic to $\Bbb{Z}$. So now we need to compute the $d_i$'s by the cellular boundary formula:
The degree of the map is $2$ when $n$ is even and $0$ when $n$ is odd. So we have groups $\Bbb{Z}$ and $2\Bbb{Z}$ and their quotient etc. It's easy to find the $\ker{d_n}/\operatorname{Im}d_{n+1}$.
But this calculation is very similar to the calculation of the homology groups of $\Bbb{R}P^n$, which is already given in the book. So is there something wrong? Is there some part I should be extra careful?
