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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Autores principales:	Barmak, J.A., Minian, E.G.
Formato:	Artículo publishedVersion
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 http://repositoriouba.sisbi.uba.ar/gsdl/cgi-bin/library.cgi?a=d&c=artiaex&d=paper_00018708_v218_n1_p87_Barmak_oai
Aporte de:	Repositorio Digital de la Universidad de Buenos Aires (UBA) de Universidad de Buenos Aires

id	I28-R145-paper_00018708_v218_n1_p87_Barmak_oai
record_format	dspace
spelling	I28-R145-paper_00018708_v218_n1_p87_Barmak_oai2020-10-19 Barmak, J.A. Minian, E.G. 2008 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. application/pdf http://hdl.handle.net/20.500.12110/paper_00018708_v218_n1_p87_Barmak info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar Adv. Math. 2008;218(1):87-104 Finite spaces Posets Simple homotopy equivalences Simple homotopy types Simplicial complexes Weak homotopy equivalences Simple homotopy types and finite spaces info:eu-repo/semantics/article info:ar-repo/semantics/artículo info:eu-repo/semantics/publishedVersion http://repositoriouba.sisbi.uba.ar/gsdl/cgi-bin/library.cgi?a=d&c=artiaex&d=paper_00018708_v218_n1_p87_Barmak_oai
institution	Universidad de Buenos Aires
institution_str	I-28
repository_str	R-145
collection	Repositorio Digital de la Universidad de Buenos Aires (UBA)
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 http://repositoriouba.sisbi.uba.ar/gsdl/cgi-bin/library.cgi?a=d&c=artiaex&d=paper_00018708_v218_n1_p87_Barmak_oai
work_keys_str_mv	AT barmakja simplehomotopytypesandfinitespaces AT minianeg simplehomotopytypesandfinitespaces
_version_	1766026465421295616

Simple homotopy types and finite spaces

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