Can $\overline{X} \in \mathbb{RP}^3,\mathbb{R}^5,\mathbb{R}^2$ cover $X \in \mathbb{RP}^2, \mathbb{RP}^5, \mathbb{RP}^2 \setminus \{P\}$

Question

The following was an exam question on my algebraic topology course:

For the following settings, determine if $\overline{X}$ can be a covering of $X$:

$\overline{X} = \mathbb{RP}^3,X = \mathbb{RP}^2$

$\overline{X} = \mathbb{R}^5, X = \mathbb{RP}^5$

$\overline{X} = \mathbb{R}^2,X = \mathbb{RP}^2 \setminus \{P\}$

I don't think I can use the injection of the fundamental group of the cover to reject any of these cases. So do I need to provide explicit coverings?

Answer 1

1 no because for a covering map $p:X\rightarrow Y$, $X$ and $Y$ must have the same dimension,

2. is not possible since $S^5$ is the universal cover of $P\mathbb{R}^5$ and $\mathbb{R}^5$ is simply connected, the universal cover is unique up to homeomorphism.

3. Possible, $P\mathbb{R}^2-P$ is covered by $S^2-\{Q,Q'\}$ and the sphere minus two points is homeomorphic via the stereographic projection to $\mathbb{R}^2-Point$ and $\mathbb{R}^2-Point$ is homeomorphic to a cylinder via polar coordinates and the cylinder is covered by $\mathbb{R}^2$.