Covering on CW complex induces homeomorphism

Question

Given a finite covering $$p: \mathbb{RP}^{2n}\longrightarrow Y$$ where $Y$ is a CW complex then $p$ is a homeomorphism.

I know that the only thing that i have to see is injectivity by the hypothesis of being a covering. I am looking just for hints and, if possible, a hint to extract what's the important property of $\mathbb{RP}^{2n}$ to generalize.

Thanks in advance

Answer 1

Hint: Consider the covering $q:S^{2n}\rightarrow RP^{2n}$. You obtain a covering $p\circ q:S^{2n}\rightarrow X$. Use the fact the Euler characteristic of the $2n$-sphere is 2, and $\chi(S^{2n})=deg(p\circ q)\chi(X)$. To determine $deg(p\circ q)$.