Studying for a qualifying exam, and came across this problem:
Let $p: S^n \to RP^n (n \geq 2)$ be the standard two-fold covering, and let $X \subset RP^n$. Prove that $p^{-1}(X)$ is path-connected if and only if $X$ is path-connected and the inclusion $i: X \hookrightarrow RP^n$ induces a surjection $i_\ast : \pi_1(X, x_0) \to \pi_1(RP^n, x_0)$ for all $x_0 \in X$.
I believe that I can show $p^{-1}(X)$ is path-connected if and only if $X$ is path-connected (follows from lifting and surjectivity of $p$). I do not see 1) If "the inclusion $i: X \hookrightarrow RP^n$ induces a surjection $i_\ast : \pi_1(X, x_0) \to \pi_1(RP^n, x_0)$ for all $x_0 \in X$" is needed for the if and only if statement and 2) how to show the surjection either. Ideas?
For any path connected $X \subset \mathbb{R}P^n$ we have that $p^{-1}(X)$ consists of at most two path components, consider the case $X = \{x_0 \}$ for a case where $X$ is path connected but $p^{-1}(X)$ isn't.
Now assume that $p^{-1}(X)$ is path connected and we show that the map $\pi_1(X, x_0) \rightarrow \pi_1(\mathbb R P ^n , x_0)$ is surjective. We will use that $\pi_1(\mathbb{R}P^n, x_0) = \mathbb Z_2$.
Let $p^{-1}(x_0) = \{s_0, s_1 \}$ then there is a path $\gamma$ from $s_0$ to $s_1$ in $p^{-1}(X)$. This becomes a loop at $x_0$, $p \circ \gamma$ in $X$ and in $\pi_1(\mathbb{R}P^n, x_0)$ it is a generator (since its lift is not a loop) so the homomorphism is in fact surjective.
If we assume that the map $\pi_1(X, x_0) \rightarrow \pi_1(\mathbb R P ^n , x_0)$ is surjective and we want to show that $p^{-1}(X)$ is path connected if $X$ is path connected we use a similar argument.
We construct a path connecting $s_0$ to $s_1$ that lies in $p^{-1}(X)$ by choosing any path $\gamma: I \rightarrow X$ that generates $\pi_1(\mathbb R P^n, x_0)$ (this exists by surjectivity of the homomorphism) and taking its lift $\gamma'$ to be the path connecting $s_0$ to $s_1$.