# Lifting homotopy in fiber bundle pairs

I'm studying the path lifting property, and I'd like to understand the case of a fiber bundle pair. Let me consider one example.

Let $p:(E,E') \rightarrow B$ be a fiber bundle pair, where the fiber pair $(F,F')$ consists of a $2$-dimensional surface $F$ and $F'\subset F$ is the union of two disjoint open subsets $U$, $V$ of $F$.

Now consider a homotopy $\Gamma:[0,1]\times[0,1]\rightarrow B$ of the form $\Gamma(t,s) = \gamma(t)$ where $\gamma$ is a curve on $B$, and let $\hat \Gamma_0:[0,1]\rightarrow E$ be a curve in $F_{\gamma(0)}$ such that $\hat \Gamma_0(0) \in U$ and $\hat \Gamma_0(1) \in V$ and lifting $s\mapsto \Gamma(0,s).$

1. Can I lift $\Gamma$ to a $\hat \Gamma:[0,1]\times[0,1]\rightarrow E$ in such a way that $\hat \Gamma(0,s)= \hat \Gamma_0(s)$ and the end-points are always inside $U$ and $V$?
2. Is it true that any other such lift is homotopic to this one via a fiber-preserving homotopy which also preserves $U$ and $V$?

I've found this topic in Spanier but I find it difficult, while Hatcher doesn't go into much detail. I'll appreciate a reference. Thank you!

• You have $\hat \Gamma_0(0) \in U$, $\hat \Gamma_0(0) \in V$ but defined $U \cap V = \emptyset$ – Birch Bryant Jun 24 '17 at 20:24
• Thanks, I'll fix the typo – user336494 Jun 24 '17 at 20:28