# lifting injective maps to injective maps in principal bundles

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!!!

• Which uniqueness? I don't need any kind of uniqueness! May 24, 2014 at 7:08
• Sorry, it was very late ^^ May 24, 2014 at 9:38
• As far as comparing the notion of fiber bundles and covering spaces, covering spaces are principal $\Gamma$-bundles, where $\Gamma$ is a subgroup of $\pi_1$ given the discrete topology.
– Neal
May 24, 2014 at 11:24
• Neal: what does your comment imply? May 24, 2014 at 13:18