Let $i \, :Y \hookrightarrow X$ be an inclusion of (nice) topological spaces, and suppose that the induced map $\pi_1(Y) \to \pi_1(X)$ is injective. Then every lifting of $i$: $$ \tilde Y \to \tilde X $$ between the universal fiber bundles of $Y$ and $X$ is injective too (moreover, the universal covering space of $Y$ is the unique one contained in $X$).
I was wondering: is there a (somehow) analogous result for principal bundles?
More generally: is there some text comparing the notions of fiber bundles and covering spaces?
I try to make the question clearer: if $p \,: \tilde X \to X$ is the universal (i.e. simply connected) covering space of a connected space $X$, and $Y \subset X$ is a $\pi_1$--injective connected subspace of $X$, then every connected component of $p^{-1}(Y)$ is a universal covering space of $Y$.
Suppose now that $p \,: E \to B$ is a principal bundle and $B' \subset B$. A statement analogous as the one above would be something like:
"Suppose that $E$ is $n+1$--connected (i.e., $\pi_i(E) = 0$ for $0 \le i \le n+1$), $B$ and $B' \subset B$ are $n$--connected, and $\pi_{n+1}(B') \to \pi_{n+1}(B)$ is injective. Then every connected component of $p^{-1}(B') \to B'$ is a $n+1$--connected principal bundle over $B'$."
I'm asking if the above statement or "something similar" is true.
mmmhhh... the statement above is obviously wrong!!!