Combinatorial fiber bundles and fragmentation of a fiberwise PL-homeomorphism

TL;DR

This paper constructs combinatorial models for PL fiber bundles and classifies their homotopy types, providing new insights into the topology of PL-manifolds and classical groups like BO(n).

Contribution

It introduces a category T(X) for combinatorial manifolds and proves a homotopy equivalence with BPL(X), enabling combinatorial models for PL fiber bundles.

Findings

BT(X) is homotopy equivalent to BPL(X)

Provides combinatorial models for PL fiber bundles

Models of real Grassmannians for small dimensions

Abstract

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.