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Abstract:

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.

Registro:

Documento: Artículo
Título:Simple homotopy types and finite spaces
Autor:Barmak, J.A.; Minian, E.G.
Filiación:Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina
Palabras clave:Finite spaces; Posets; Simple homotopy equivalences; Simple homotopy types; Simplicial complexes; Weak homotopy equivalences
Año:2008
Volumen:218
Número:1
Página de inicio:87
Página de fin:104
DOI: http://dx.doi.org/10.1016/j.aim.2007.11.019
Título revista:Advances in Mathematics
Título revista abreviado:Adv. Math.
ISSN:00018708
PDF:https://bibliotecadigital.exactas.uba.ar/download/paper/paper_00018708_v218_n1_p87_Barmak.pdf
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00018708_v218_n1_p87_Barmak

Referencias:

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Citas:

---------- APA ----------
Barmak, J.A. & Minian, E.G. (2008) . Simple homotopy types and finite spaces. Advances in Mathematics, 218(1), 87-104.
http://dx.doi.org/10.1016/j.aim.2007.11.019
---------- CHICAGO ----------
Barmak, J.A., Minian, E.G. "Simple homotopy types and finite spaces" . Advances in Mathematics 218, no. 1 (2008) : 87-104.
http://dx.doi.org/10.1016/j.aim.2007.11.019
---------- MLA ----------
Barmak, J.A., Minian, E.G. "Simple homotopy types and finite spaces" . Advances in Mathematics, vol. 218, no. 1, 2008, pp. 87-104.
http://dx.doi.org/10.1016/j.aim.2007.11.019
---------- VANCOUVER ----------
Barmak, J.A., Minian, E.G. Simple homotopy types and finite spaces. Adv. Math. 2008;218(1):87-104.
http://dx.doi.org/10.1016/j.aim.2007.11.019