TITLE: Singular homology of $\mathbb{R}P^{2}$
QUESTION [8 upvotes]: I am doing algebraic topology course and I'm trying to compute the singular homology for some spaces.
However I am stuck doing that for $\mathbb{R}P^{2}$.
Compute the singular homology of $\mathbb{R}P^{2}$ (the real projective plane).

REPLY [4 votes]: $\mathbb{R}P^2$ is a CW complex with one 0-cell $v$, one 1-cell $a$, one 2-cell $e$ which is attached via the word $a^2$. Cellular homology comes from the chain complex $$0\rightarrow H_2(e,a)\rightarrow H_1(a,v){\rightarrow }H_0(v,\emptyset)\rightarrow 0$$ where $H_i$ are the $i$-th singular homology groups of consecutive pairs of skeletons for the CW structure. Each of these groups in this complex is just a direct sum of copies of $\mathbb{Z}$ with a generator for each $i$-cell. In this case, $$0\rightarrow\mathbb{Z}\rightarrow\mathbb{Z}\rightarrow\mathbb{Z}\rightarrow 0.$$ The boundary maps are computed via the degree of the attaching maps for each cell that generates a $\mathbb{Z}$. In this case the only intersting boundary map is $d_2:[e]\mapsto 2[a]$ where 2 is the degree of the attaching map of $e$ to $a$. All other boundary maps are $0$. So the cellular homology groups (which are the same a singular homology groups) are $H_n(\mathbb{R}P^2)=0$ for $n>1$, $H_1(\mathbb{R}P^2)=\mathbb{Z}_2$, $H_0(\mathbb{R}P^2)=\mathbb{Z}.$  
Cellular homology works very efficiently for computing homology groups of most  familiar CW spaces. i.e., $\mathbb{C}P^n, \mathbb{R}P^n$, genus $g$ surfaces, etc. Hatcher has a lot of examples in the middle of his Homology chapter, and it's very readable.