Combinatorial fiber bundles and fragmentation of a fiberwise PL homeomorphism

Abstract

With a compact PL manifold X we associate a category $\mathfrak{T}(X)$. The objects of $\mathfrak{T}(X)$ are all combinatorial manifolds of type X, and morphisms are combinatorial assemblies. We prove that the homotopy equivalence

$$B\mathfrak{T}(X) \approx BPL(X)$$

holds, where PL(X) is the simplicial group of PL homeomorphisms. Thus the space $B\mathfrak{T}(X)$ 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 $\mathfrak{T}(X)$-coloring of some triangulation K of B. The vertices of K are colored by objects of $\mathfrak{T}(X)$, 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: $B\mathfrak{T}(S^{n - 1} ) \approx BO(n)$ for n = 1, 2, 3, 4. The result is proved in a sequence of results on similar models of BPL (X). Special attention is paid to the main noncompact case X = ℝⁿ 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.