COMBINATORIAL FIBER BUNDLES AND FRAGMENTATION OF A FIBERWISE PL HOMEOMORPHISM

摘要

With a compact PL manifold X we associate a category S{X). The objects of S(X) are all combinatorial manifolds of type X, and morphisms are combinatorial assemblies. We prove that the homotopy equivalence BS(X) ≈ BPL (X) holds, where PL (X) is the simplicial group of PL homeomorphisms. Thus the space BS(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 S(X) -coloring of some triangulation K of B. The vertices of K are colored by objects of S(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: BS(S~(n-1)) ≈ BO(n) for n = 1,2,3,4. The result is proved in a sequence of results on similar models of BPL (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. Bibliography: 34 titles.