# Fiberwise Characteristic Classes

I recently became interested in a problem of detecting the (non)existence of a section for a bundle where the fibers are not constant. Let $$I$$ be the unit interval and let $$F_t$$, $$t \in [0,1]$$ be time-varying fibers. The (standard) picture is this:

$$\require{AMScd} \begin{CD} \cup_{t \in I} F_t @>>> \cup_{t \in I} I \times F_t\\ & @VVV \\ && I \end{CD}$$

My first thought was to use characteristic classes, so I was wondering if anyone could tell me a little bit about "time-varying characteristic classes" or direct me to an appropriate reference.

• The usual definition of fiber bundle forces the fibers to be homeomorphic (at least over connected bases). What exactly do you mean by time varying fiber bundle? – Jason DeVito Sep 11 at 11:40