Let $A\overset{\alpha}{\rightarrow}B$ be a Hurewicz fibration. Any Hurewicz connection defines parallel transport along curves in the base. In general, such parallel transport maps $\alpha^{-1}(b)\to \alpha^{-1}(b^\prime)$ are continuous homotopy equivalences, but they need not be homeomorphisms.

For instance, consider the fibration given by projecting the standard 2-simplex onto the $x$-axis. For a connection we can lift a path in the $x$-axis by composing it with the embedding of the $x$-axis in the 2-simplex as a "rotation" of the $x$-axis with $(0,1)$ fixed.

The pleasant thing is that a Hurewicz fibration admits a connection-independent functor $\pi_1B\to \mathsf{hTop}$ taking a homotopy class of paths to the homotopy class of some parallel transport along it.

I am trying to understand when a particular Hurewicz connection might induce a parallel transport functor $\mathsf{Paths}(B)\to \mathsf{Top}$. Here $\mathsf{Paths}(B)$ is the groupoid whose objects are the points of $B$ and whose arrows are continuous maps from an interval of the form $[0,\ell]$ to $B$. Composition of paths is defined to be associative.

Such a transport functor would force the fibers to be homeomorphic, so I might as well assume the fibration is a fiber bundle. Still though, I don't what might force the parallel transports of arbitrary Hurewicz connection on a fiber bundle to behave functorially (even to just be homeomorphisms).

The same question remains if we restrict to loops and replace the path groupoid with Moore loops.

Since we assume $A\overset{\alpha}{\rightarrow}B$ is a fiber bundle, pulling back along a curve gives a fiber bundle over an interval, which must be trivial, but I don't really see how this helps. (In the smooth theory, parallel transports are diffeomorphisms because they come from the flow of a vector field. I don't see an analogue here.)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.