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Abstract:

The classical way to study a finite poset (X, ≤) using topology is by means of the simplicial complex Δ X of its nonempty chains. There is also an alternative approach, regarding X as a finite topological space. In this article we introduce new constructions for studying X topologically: inspired by a classical paper of Dowker (Ann Math 56:84-95, 1952), we define the simplicial complexes K X and L X associated to the relation ≤. In many cases these polyhedra have the same homotopy type as the order complex Δ X . We give a complete characterization of the simplicial complexes that are the K or L-complexes of some finite poset and prove that K X and L X are topologically equivalent to the smaller complexes K′ X , L′ X induced by the relation ≤. More precisely, we prove that K X (resp. L X ) simplicially collapses to K′ X (resp. L′ X ). The paper concludes with a result that relates the K-complexes of two posets X, Y with closed relations R ⊂ X × Y. © 2010 Springer Science+Business Media B.V.

Registro:

Documento: Artículo
Título:The Geometry of Relations
Autor:Minian, E.G.
Filiación:Departamento de Matemática FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina
Palabras clave:Collapses; Finite spaces; Nerves; Posets; Relations; Simplicial complexes
Año:2010
Volumen:27
Número:2
Página de inicio:213
Página de fin:224
DOI: http://dx.doi.org/10.1007/s11083-010-9146-4
Título revista:Order
Título revista abreviado:Order
ISSN:01678094
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_01678094_v27_n2_p213_Minian

Referencias:

  • Alexandroff, P.S., Diskrete Räume (1937) MathematiceskiiSbornik (N.S.), 2, pp. 501-518
  • Bak, A., Brown, R., Minian, E.G., Porter, T., Global actions, groupoid atlases and applications (2006) J. Homotopy Relat. Struct., 1, pp. 101-167
  • Barmak, J.A., Minian, E.G., 2-Dimension from the topological viewpoint (2007) Order, 24, pp. 49-58
  • Barmak, J.A., Minian, E.G., Minimal finite models (2007) J. Homotopy Relat. Struct., 2, pp. 127-140
  • Barmak, J.A., Minian, E.G., Simple homotopy types and finite spaces (2008) Adv. Math., 218, pp. 87-104
  • Björner, A., Topological methods (1995) Handbook of Combinatorics, , R. Graham, M. Grötschel, L. Lovász (Eds.), New York: Elsevier
  • Cohen, M.M., (1970) A Course in Simple Homotopy Theory, , New York: Springer-Verlag
  • del, H.M., Minian, E.G., Classical invariants for global actions and groupoid atlases (2008) Appl. Categ. Struct., 16 (6), pp. 689-721
  • Dowker, C.H., Homology groups of relations (1952) Ann. Math., 56, pp. 84-95
  • Kozlov, D., (2008) Combinatorial Algebraic Topology, 21. , Algorithms and Computation in Mathematics, Berlin: Springer
  • May, J.P., Finite topological spaces (2003) Notes for REU, , http://www.math.uchicago.edu/may/MISCMaster.html, Available at
  • McCord, M.C., Singular homology groups and homotopy groups of finite topological spaces (1966) Duke Math. J., 33, pp. 465-474
  • McCord, M.C., Homotopy type comparison of a space with complexes associated with its open covers (1967) Proc. Am. Math. Soc., 18 (4), pp. 705-708
  • Milnor, J., Whitehead Torsion (1966) Bull. Amer. Math. Soc., 72, pp. 358-426
  • Quillen, D., Higher algebraic K-theory I (1973) Lect. Notes Math., 341, pp. 85-147
  • Quillen, D., Homotopy properties of the poset of non-trivial p-subgroups of a group (1978) Adv. Math., 28, pp. 101-128
  • Segal, G., Classifying spaces and spectral sequences (1968) Inst. Hautes Etudes Sci. Publ. Math., 34, pp. 105-112
  • Siebenmann, L.C., Infinite simple homotopy types (1970) Indag. Math., 32, pp. 479-495
  • Spanier, E., (1966) Algebraic Topology, , New York: McGraw Hill
  • Stanley, R., Enumerative Combinatorics (1997) Cambrdige Studies in Advanced Mathematics, 1, p. 49. , Cambridge: Cambridge University Press
  • Stong, R.E., Finite topological spaces (1966) Trans. Am. Math. Soc., 123, pp. 325-340
  • Whitehead, J.H.C., Simplicial spaces, nuclei and m-groups (1939) Proc. Lond. Math. Soc., 45, pp. 243-327
  • Whitehead, J.H.C., On incidence matrices, nuclei and homotopy types (1941) Ann. Math., 42, pp. 1197-1239
  • Whitehead, J.H.C., Simple homotopy types (1950) Am. J. Math., 72, pp. 1-57
  • Zeeman, E.C., On the dunce hat (1964) Topology, 2, pp. 341-358

Citas:

---------- APA ----------
(2010) . The Geometry of Relations. Order, 27(2), 213-224.
http://dx.doi.org/10.1007/s11083-010-9146-4
---------- CHICAGO ----------
Minian, E.G. "The Geometry of Relations" . Order 27, no. 2 (2010) : 213-224.
http://dx.doi.org/10.1007/s11083-010-9146-4
---------- MLA ----------
Minian, E.G. "The Geometry of Relations" . Order, vol. 27, no. 2, 2010, pp. 213-224.
http://dx.doi.org/10.1007/s11083-010-9146-4
---------- VANCOUVER ----------
Minian, E.G. The Geometry of Relations. Order. 2010;27(2):213-224.
http://dx.doi.org/10.1007/s11083-010-9146-4