TY - JOUR

T1 - Combinatorial fiber bundles and fragmentation of a fiberwise PL homeomorphism

AU - Mnëv, N. E.

PY - 2007/12/1

Y1 - 2007/12/1

N2 - 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 B {T}(X) BPL(X) holds, where PL(X) is the simplicial group of PL homeomorphisms. Thus the space B{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 {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: B {T}(Sn - 1) 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. © 2007 Springer Science+Business Media, Inc.

AB - 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 B {T}(X) BPL(X) holds, where PL(X) is the simplicial group of PL homeomorphisms. Thus the space B{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 {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: B {T}(Sn - 1) 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. © 2007 Springer Science+Business Media, Inc.

UR - http://www.scopus.com/inward/record.url?scp=36148953859&partnerID=8YFLogxK

U2 - 10.1007/s10958-007-0537-z

DO - 10.1007/s10958-007-0537-z

M3 - Article

AN - SCOPUS:36148953859

VL - 147

SP - 7155

EP - 7217

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 6

ER -