The classification of PL fibrations.

Steinberger, Mark

doi:10.1307/mmj/1029003286

Cited by 8 publications

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“…Here B * is the classifying space functor, which coincides in the case of posets with the geometric realization of the order complex, in the case of categories -with Milnor geometric realization of the nerve and in the case of simplicial group with the classifying space for principal bundles. These results can be viewed as a purely combinatorial variation of some constructions from [Lev89], [Hat75], [Ste86]. Also this proves the PL double for MacPherson's conjecture on modeling the real Supported by RFBR grant 05-01-00899 and S.S. grant 4329.2006.1.…”

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confidence: 55%

“…By abstract nonsense, to prove (1) is the same as to establish a functorial one-to-one correspondence between the isomorphism classes of PL fiber bundles on L with fiber X and the concordance classes of R(X)-colorings of L. We mention how we would like, but cannot, establish such a correspondence. This speculation is borrowed from [15]. To any R(X)-coloring of L we can apply the construction Hocolim, and its geometric realization will produce a triangulated fiber bundle with fiber X.…”

Section: Abstract Manifoldsmentioning

confidence: 99%

“…We will menton how we would like but cannot establish such a correspondence. This speculation is borrowed from [Ste86]. To any R(X)-coloring of L we can apply the construction Hocolim, and its geometric realization will produce a triangulated fiber bundle with fiber X.…”

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confidence: 99%

“…To any R(X)-coloring of L we can apply the construction Hocolim, and its geometric realization will produce a triangulated fiber bundle with fiber X. On the other side one can triangulate any fiber bundle with the base L. To any triangulated fiber bundle J with base L and fiber X one can canonically associate ( [Hat75], [Ste86]) some R(X)-coloring of the first barycentric subdivision of the base of J. We call this construction by Hocolim −1 .…”

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A Note on the Tangent Bundle and Gauss Functor of Posets and Manifolds

Mnëv

2014

J Math Sci

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confidence: 55%

Section: Abstract Manifoldsmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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A Note on the Tangent Bundle and Gauss Functor of Posets and Manifolds

Mnëv

2014

J Math Sci

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“…Let us describe how we would like to, but cannot, prove such a theorem. This speculation is borrowed from [32].…”

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Combinatorial fiber bundles and fragmentation of a fiberwise PL-homeomorphism

Mnev

2007

Preprint

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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 equivalenceholds, 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 ) ≈ 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 = 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.

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Geometric modules and algebraicK-homology theory

Anderson¹,

Munkholm²

1990

K-Theory

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The classification of PL fibrations.

Cited by 8 publications

References 0 publications

A Note on the Tangent Bundle and Gauss Functor of Posets and Manifolds

A Note on the Tangent Bundle and Gauss Functor of Posets and Manifolds

Combinatorial fiber bundles and fragmentation of a fiberwise PL-homeomorphism

Geometric modules and algebraicK-homology theory

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