Simplicial approximation of a fibration

Question

I am interested in the simplicial approximation of Serre or Hurewicz fibrations (or even fibre bundles). Let's assume $E$ and $B$ are finite simplicial complexes (or their associated geometric realizations) (therefore compact spaces) and $p\colon E \to B$ is a fibration (in the topological sense, I mean, the homotopy lifting property is satisfied).

Are there any sufficient conditions which guarantee that a simplicial approximation of $p$, $\widetilde{p}\colon E\to B$, is also a fibration? or at least that all the fibres of $\widetilde{p}$ have the same homotopy type and the same homotopy type of the fibers of $p$ (in the case of Hurewicz fibrations)? And in the case of fiber bundles?

I am both interested in arguments or references where you think some information about these topics could be provided. I am also interested in counterexamples which show under which conditions what I ask is not possible.

Just to clarify: my definition of simplicial map that for every simplex of $E$, it takes its vertices to the vertices of a simplex of $B$, and it is affine on each simplex. (See for example Bredon's Topology and Geometry.)

Thanks in advance and any help would be appreciated.

Answer 1

This is interesting a question. If E and B are smooth triangulated manifolds and if p is a smooth bundle map, then you could begin with a subdivision and a "jiggling" to put the triangulation in "general position" wrt the fibres of p (Thurston, Commentarii 1974). Anyway, may I point out the criteria for fibrations and bundles in an old paper of mine, that could maybe help you: G. Meigniez, "Submersions, fibrations and bundles", Trans. Amer. Math. Soc. 354 (2002), 3771-3787, [link] (http://web.univ-ubs.fr/lmba/meigniez/docu/preprints/sfb.pdf)