Combinatorial PL fiber bundles and fragmentation of a fiberwise homeomorphism

Among other things, we prove that for any compact PL-manifold $X$ there is a homotopy equivalence $BPL(X)\approx BT(X)$, where $T(X)$ is the category of abstract aggregations of triangulations of $X$. As a result, we get a functorial pure combinatorial models for PL fiber bundles. Special attention is paid to the case $X=\mathbb R^n$ and the combinatorial model of the Gauss map of a combinatorial manifold. The key trick which makes the proof possible is a collection of lemmas describing the fragmentation of a fiberwise homeomorphism.