  With a compact PL manifold X we associate a category T(X). The objects of
T(X) are all combinatorial manifolds of type X, and morphisms are combinatorial
assemblies. We prove that the homotopy equivalence BT (X) \approx BPL(X) holds,
where PL(X) is the simplicial group of PL-homeomorphisms. Thus the space BT(X)
is a canonical countable (as a CW-complex) model of BPL(X). As a result, we
obtain functorial pure combinatorial models for PL fiber bundles with fiber X
and a PL polyhedron B as the base. Such a model looks like a T(X)-coloring of
some triangulation K of B. The vertices of K are colored by objects of T(X) and
the arcs are colored by morphisms in such a way that the diagram arising from
the 2-skeleton of K is commutative. Comparing with the classical results of
geometric topology, we obtain combinatorial models of the real Grassmannian in
small dimensions: BT(S^{n-1}) \approx BO(n) for n=1,2,3,4. The result is proved
in a sequence of results on similar models of B\PL(X). Special attention is
paid to the main noncompact case X=R^n and to the tangent bundle and Gauss
functor of a combinatorial manifold. The trick that makes the proof possible is
a collection of lemmas on "fragmentation of a fiberwise homeomorphism", a
generalization of the folklore lemma on fragmentation of an isotopy.

Geometric Topology;Combinatorics;Category Theory