Simple homotopy types and finite spaces

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We present a new approach to simple homotopy theory of polyhedra using finite topological spaces. We define the concept of collapse of a finite space and prove that this new notion corresponds exactly to the concept of a simplicial collapse. More precisely, we show that a collapse X ↘ Y of finite sp...

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Detalles Bibliográficos
Autores principales: Barmak, J.A., Minian, E.G.
Formato: Artículo publishedVersion
Lenguaje:Inglés
Publicado: 2008
Materias:
Finite spaces
Posets
Simple homotopy equivalences
Simple homotopy types
Simplicial complexes
Weak homotopy equivalences
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_00018708_v218_n1_p87_Barmak
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id paperaa:paper_00018708_v218_n1_p87_Barmak
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spelling paperaa:paper_00018708_v218_n1_p87_Barmak2023-06-12T16:39:27Z Simple homotopy types and finite spaces Adv. Math. 2008;218(1):87-104 Barmak, J.A. Minian, E.G. Finite spaces Posets Simple homotopy equivalences Simple homotopy types Simplicial complexes Weak homotopy equivalences We present a new approach to simple homotopy theory of polyhedra using finite topological spaces. We define the concept of collapse of a finite space and prove that this new notion corresponds exactly to the concept of a simplicial collapse. More precisely, we show that a collapse X ↘ Y of finite spaces induces a simplicial collapse K (X) ↘ K (Y) of their associated simplicial complexes. Moreover, a simplicial collapse K ↘ L induces a collapse X (K) ↘ X (L) of the associated finite spaces. This establishes a one-to-one correspondence between simple homotopy types of finite simplicial complexes and simple equivalence classes of finite spaces. We also prove a similar result for maps: We give a complete characterization of the class of maps between finite spaces which induce simple homotopy equivalences between the associated polyhedra. This class describes all maps coming from simple homotopy equivalences at the level of complexes. The advantage of this theory is that the elementary move of finite spaces is much simpler than the elementary move of simplicial complexes: It consists of removing (or adding) just a single point of the space. © 2007 Elsevier Inc. All rights reserved. Fil:Barmak, J.A. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Minian, E.G. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2008 info:eu-repo/semantics/article info:ar-repo/semantics/artículo info:eu-repo/semantics/publishedVersion application/pdf eng info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00018708_v218_n1_p87_Barmak
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
language Inglés
orig_language_str_mv eng
topic Finite spaces
Posets
Simple homotopy equivalences
Simple homotopy types
Simplicial complexes
Weak homotopy equivalences
spellingShingle Finite spaces
Posets
Simple homotopy equivalences
Simple homotopy types
Simplicial complexes
Weak homotopy equivalences
Barmak, J.A.
Minian, E.G.
Simple homotopy types and finite spaces
topic_facet Finite spaces
Posets
Simple homotopy equivalences
Simple homotopy types
Simplicial complexes
Weak homotopy equivalences
description We present a new approach to simple homotopy theory of polyhedra using finite topological spaces. We define the concept of collapse of a finite space and prove that this new notion corresponds exactly to the concept of a simplicial collapse. More precisely, we show that a collapse X ↘ Y of finite spaces induces a simplicial collapse K (X) ↘ K (Y) of their associated simplicial complexes. Moreover, a simplicial collapse K ↘ L induces a collapse X (K) ↘ X (L) of the associated finite spaces. This establishes a one-to-one correspondence between simple homotopy types of finite simplicial complexes and simple equivalence classes of finite spaces. We also prove a similar result for maps: We give a complete characterization of the class of maps between finite spaces which induce simple homotopy equivalences between the associated polyhedra. This class describes all maps coming from simple homotopy equivalences at the level of complexes. The advantage of this theory is that the elementary move of finite spaces is much simpler than the elementary move of simplicial complexes: It consists of removing (or adding) just a single point of the space. © 2007 Elsevier Inc. All rights reserved.
format Artículo
Artículo
publishedVersion
author Barmak, J.A.
Minian, E.G.
author_facet Barmak, J.A.
Minian, E.G.
author_sort Barmak, J.A.
title Simple homotopy types and finite spaces
title_short Simple homotopy types and finite spaces
title_full Simple homotopy types and finite spaces
title_fullStr Simple homotopy types and finite spaces
title_full_unstemmed Simple homotopy types and finite spaces
title_sort simple homotopy types and finite spaces
publishDate 2008
url http://hdl.handle.net/20.500.12110/paper_00018708_v218_n1_p87_Barmak
work_keys_str_mv AT barmakja simplehomotopytypesandfinitespaces
AT minianeg simplehomotopytypesandfinitespaces
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