Combinatorial fiber bundles and fragmentation of a fiberwise PL-homeomorphism

Details Combinatorial fiber bundles and fragmentation of a fiberwise PL-homeomorphism Combinatorial fiber bundles and fragmentation of a fiberwise PL-homeomorphism

2007-08-30

arxiv.org/abs/0708.4039v3

Zapiski Nauchnyh Seminarov POMI, v344, p 56-173, 2007 (Russian)

Mathematics

Geometric Topology

104 pages 21 figures, minor English update

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Combinatorial fiber bundles and fragmentation of a fiberwise PL-homeomorphism does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Combinatorial fiber bundles and fragmentation of a fiberwise PL-homeomorphism, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Combinatorial fiber bundles and fragmentation of a fiberwise PL-homeomorphism will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-205407