Why the lens space L(2,1) is homeomorphic to $\mathbb{R}P^3$? According to one definition of lens space $L(p,q)$, which is gluing two solid tori with a map $h:T^2_1 \rightarrow T^2_2$. And $h(m_1)=pl_2+qm_2$, $l_i$ means longitude and $m_i$ means meridian of the boundary torus. I cannot understand why $L(2,1)$ is homeomorphic to $\mathbb{R}P^3$. Can someone give me some hints? Thank you.

If the definition of lens space $L(p,q)$ is given as two solid tori glued together, then probably you know that $S^3$ is the result of two solid tori glued together as well (meridian to longitude, and longitude to meridian), denote them $T_1$ and $T_2$.
Say $T_1$ is centered at the origin of $\mathbb{R}^3$ (and think $S^3$ as the compactification of $\mathbb{R}^3$), and consider $\mathbb{R}P^3$ as $S^3$ with antipodal points identified (~). We can see that $T_1/\sim$ is still a solid torus (there is a twist in the second step): 
Consider a longitude on $T_1$, and we can see as below that after taking the quotient it becomes a curve that goes once around the meridian and twice around the longitude, let's call this curve $\alpha$: 
Now think about $T_2$ and do the similar thing to a meridian curve (and the result is still a meridian).
Remember $S^3$ is the result by gluing a meridian of $T_2$ to a longitude of $T_1$, thus $\mathbb{R}P^3$ is the result by gluing meridian of a solid torus to the curve $\alpha$ on the other solid torus, but that is exactly how your $L(2,1)$ is defined.

I personally found this an easier way to understand why they are the same with the 'gluing' definition, because this way you naturally have two solid torus from $\mathbb{R}P^3$ without building/finding them.
