# Projective space, explicit descriptions of maps.

Consider $\mathbb{P}_\mathbb{R}^2$, i.e. the $\mathbb{R}$-projective plane. I have two questions.

My first question is, what is an explicit description of the isomorphism between $H_*^{\text{cellular}}(\mathbb{P}_\mathbb{R}^2)$ and $H_*^{\text{simplicial}}(\mathbb{P}_\mathbb{R}^2)$?

My second question is, what is an explicit description of the abelianization map from $\pi_1(\mathbb{P}_\mathbb{R}^2)$ to $H_1(\mathbb{P}_\mathbb{R}^2)$?

• Hi @DanielMcLaury, thanks for the answer! I'm aware of your remarks on homology in general, but I'm interested in a very explicit description in the case of $\mathbb{P}_\mathbb{R}^2$ of the isomorphism between cellular and simplicial homology here. – user387384 Nov 8 '16 at 18:03
• Well, of course that depends on explicitly saying which of the infinitely many possible cellular structures and simplicial structures you've chosen to endow $\mathbb{P}^2$ with. For instance, you could take both of them to be the same, at which point the map on homology is just induced by the identity map on chains. – Daniel McLaury Nov 8 '16 at 18:06