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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
Article
  • 44 Downloads

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.

Keywords

Simplicial Complex Graph System Trivial Bundle Minimal Base Ball Complex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

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

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