Given a fibration (or a fiber bundle or whatever might make my question true) $f\colon E\rightarrow B$ and basepoints $e$ in $E$ and $b$ in $B$, such that $f$ preserves them.

By the homotopy lifting property, I can lift every path $\gamma\colon I\rightarrow B$ starting from $b$ to a path in $E$ starting from $e$. Does this work in families, i.e. can I lift continuously varying paths in $B$ to continuously varying paths in $E$?

Let me make this into a precise question:

The restriction map $$Maps_*(I,B)\rightarrow Maps_*(I,E)$$ is surjective, where the path spaces consists of path starting at $b$ or $e$ respectively. Hence, this map has a set-theoretical section $$s\colon Maps_*(I,E)\rightarrow Maps_*(I,B)$$ and I want to know whether I can choose this section continuously and if not, how strong this assumption is, i.e. whether it will be fulfilled in most cases.

  • $\begingroup$ You should specify the topology on $Maps(I, \cdot)$ that you are using. $\endgroup$ – Moishe Kohan May 5 '16 at 15:13
  • $\begingroup$ The standard one (compact open or the compactly generated refinement of it, depending which category of spaces you prefer). $\endgroup$ – DaleCooper May 6 '16 at 7:15
  • $\begingroup$ Already just asking for a continuous section of $E\to B$ implies every homotopy group of $B$ is canonically a quotient of one of $E$. For instance if $E$'is the path space of $B$ itself, such a section will only exist if $B$ is contractible. $\endgroup$ – Kevin Carlson May 8 '16 at 17:16

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