Journal of Mathematical Sciences

, Volume 147, Issue 6, pp 7155–7217 | Cite as

Combinatorial fiber bundles and fragmentation of a fiberwise PL homeomorphism

  • N. E. Mnëv


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.


Simplicial Complex Graph System Trivial Bundle Minimal Base Ball Complex 
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© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.St.Petersburg Department of the Steklov Mathematical InstituteSt. PetersburgRussia

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