Simplicial fibrations

We undertake a systematic study of fibrations in the setting of abstract simplicial complexes, where the concept of “homotopy” has been replaced by that of “contiguity”. Then, a fibration will be a simplicial map satisfying the “contiguity lifting property”. This definition turns out to be equivalen...

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Authors: Fernández Ternero, Desamparados, García Calcines, José Manuel, Macías Virgós, Enrique, Vilches Alarcón, José Antonio
Format: article
Status:Versión aceptada para publicación
Publication Date:2021
Country:España
Institution:Universidad de Sevilla (US)
Repository:idUS. Depósito de Investigación de la Universidad de Sevilla
OAI Identifier:oai:idus.us.es:11441/167711
Online Access:https://hdl.handle.net/11441/167711
https://doi.org/10.1007/s13398-020-00966-5
Access Level:Open access
Keyword:Simplicial complexes
Contiguous simplicial maps
Fibrations
LS-category
Topological complexity
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spelling Simplicial fibrationsFernández Ternero, DesamparadosGarcía Calcines, José ManuelMacías Virgós, EnriqueVilches Alarcón, José AntonioSimplicial complexesContiguous simplicial mapsFibrationsLS-categoryTopological complexityWe undertake a systematic study of fibrations in the setting of abstract simplicial complexes, where the concept of “homotopy” has been replaced by that of “contiguity”. Then, a fibration will be a simplicial map satisfying the “contiguity lifting property”. This definition turns out to be equivalent to that introduced by Minian, established in terms of a cylinder construction . This allows us to prove several properties of simplicial fibrations which are analogous to the classical ones in the topological setting, for instance: all the fibers of a fibration with connected base have the same strong homotopy type and any fibration with a strongly collapsible base is fibrewise trivial. We also introduce the concept of “simplicial finite-fibration”, that is, a simplicial map which has the contiguity lifting property only for finite complexes. Then, we prove that the path fibration is a finite-fibration, where is the simplicial complex of Moore paths introduced by Grandis. This result allows us to prove that any simplicial map factors through a finite-fibration, up to a P-homotopy equivalence. Moreover, we prove a simplicial version of a Varadarajan result for fibrations, relating the LS-category of the total space, the base and the generic fiber. Finally, we introduce a definition of “Švarc genus” of a simplicial map and we are able to compare the Švarc genus of path fibrations with the notions of simplicial LS-category and simplicial topological complexity introduced by the authors in several previous papers.SpringerGeometría y TopologíaFQM326: Geometria Diferencial y Teoria de Lie2021info:eu-repo/semantics/articleinfo:eu-repo/semantics/acceptedVersionapplication/pdfapplication/pdfhttps://hdl.handle.net/11441/167711https://doi.org/10.1007/s13398-020-00966-5reponame:idUS. Depósito de Investigación de la Universidad de Sevillainstname:Universidad de Sevilla (US)InglésRevista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 115 (2), 54.https://doi.org/10.1007/s13398-020-00966-5info:eu-repo/semantics/openAccessoai:idus.us.es:11441/1677112026-06-17T12:51:07Z
dc.title.none.fl_str_mv Simplicial fibrations
title Simplicial fibrations
spellingShingle Simplicial fibrations
Fernández Ternero, Desamparados
Simplicial complexes
Contiguous simplicial maps
Fibrations
LS-category
Topological complexity
title_short Simplicial fibrations
title_full Simplicial fibrations
title_fullStr Simplicial fibrations
title_full_unstemmed Simplicial fibrations
title_sort Simplicial fibrations
dc.creator.none.fl_str_mv Fernández Ternero, Desamparados
García Calcines, José Manuel
Macías Virgós, Enrique
Vilches Alarcón, José Antonio
author Fernández Ternero, Desamparados
author_facet Fernández Ternero, Desamparados
García Calcines, José Manuel
Macías Virgós, Enrique
Vilches Alarcón, José Antonio
author_role author
author2 García Calcines, José Manuel
Macías Virgós, Enrique
Vilches Alarcón, José Antonio
author2_role author
author
author
dc.contributor.none.fl_str_mv Geometría y Topología
FQM326: Geometria Diferencial y Teoria de Lie
dc.subject.none.fl_str_mv Simplicial complexes
Contiguous simplicial maps
Fibrations
LS-category
Topological complexity
topic Simplicial complexes
Contiguous simplicial maps
Fibrations
LS-category
Topological complexity
description We undertake a systematic study of fibrations in the setting of abstract simplicial complexes, where the concept of “homotopy” has been replaced by that of “contiguity”. Then, a fibration will be a simplicial map satisfying the “contiguity lifting property”. This definition turns out to be equivalent to that introduced by Minian, established in terms of a cylinder construction . This allows us to prove several properties of simplicial fibrations which are analogous to the classical ones in the topological setting, for instance: all the fibers of a fibration with connected base have the same strong homotopy type and any fibration with a strongly collapsible base is fibrewise trivial. We also introduce the concept of “simplicial finite-fibration”, that is, a simplicial map which has the contiguity lifting property only for finite complexes. Then, we prove that the path fibration is a finite-fibration, where is the simplicial complex of Moore paths introduced by Grandis. This result allows us to prove that any simplicial map factors through a finite-fibration, up to a P-homotopy equivalence. Moreover, we prove a simplicial version of a Varadarajan result for fibrations, relating the LS-category of the total space, the base and the generic fiber. Finally, we introduce a definition of “Švarc genus” of a simplicial map and we are able to compare the Švarc genus of path fibrations with the notions of simplicial LS-category and simplicial topological complexity introduced by the authors in several previous papers.
publishDate 2021
dc.date.none.fl_str_mv 2021
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/acceptedVersion
format article
status_str acceptedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/11441/167711
https://doi.org/10.1007/s13398-020-00966-5
url https://hdl.handle.net/11441/167711
https://doi.org/10.1007/s13398-020-00966-5
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 115 (2), 54.
https://doi.org/10.1007/s13398-020-00966-5
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Springer
publisher.none.fl_str_mv Springer
dc.source.none.fl_str_mv reponame:idUS. Depósito de Investigación de la Universidad de Sevilla
instname:Universidad de Sevilla (US)
instname_str Universidad de Sevilla (US)
reponame_str idUS. Depósito de Investigación de la Universidad de Sevilla
collection idUS. Depósito de Investigación de la Universidad de Sevilla
repository.name.fl_str_mv
repository.mail.fl_str_mv
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