Covering of a submanifold

Question

Let $f:\tilde{M}\to M$ be a $d$-fold covering map and $N\subset M$ be a codimension 2 submanifold. What is the number of path-connected components of $f^{-1}(N)$ (assuming $N$ is connected)?

Answer 1

It seems you want to show that $N' = f^{-1}(N)$ has $d$ path components. But $N'$ may have only one. Example:

$M = \tilde{M} = S^1 \times S^2, f(z,\xi) = (z^d, \xi)$ (i.e. $f$ is the product of the standard $d$-fold covering of $S^1$ and the covering $id : S^2 \to S^2$).

Then $N = S^1 \times \{ a \}$ is a connected codimension $2$ submanifold such that $f^{-1}(N) = N$.

Another example is $M = \mathbb{RP}^3$ = $3$-dimensional real projective space with universal cover $p : S^3 \to \mathbb{RP}^3$ which has two sheets. The circle $S = S^1 \times \{(0,0) \} \subset S^3 \subset \mathbb{R}^4$ is mapped by $p$ to $p(S^1) = S /\sim$, where $\xi \sim \xi'$ if $\xi = \pm \xi'$. Therefore $p(S) \approx \mathbb{RP}^1 \approx S^1$. $p(S)$ is a a connected codimension $2$ submanifold of $\mathbb{RP}^3$ such that $p^{-1}(p(S)) = S$.