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Abstract:

We introduce the theory of strong homotopy types of simplicial complexes. Similarly to classical simple homotopy theory, the strong homotopy types can be described by elementary moves. An elementary move in this setting is called a strong collapse and it is a particular kind of simplicial collapse. The advantage of using strong collapses is the existence and uniqueness of cores and their relationship with the nerves of the complexes. From this theory we derive new results for studying simplicial collapsibility with a different point of view. We analyze vertex-transitive simplicial G-actions and prove a particular case of the Evasiveness conjecture for simplicial complexes. Moreover, we reduce the general conjecture to the class of minimal complexes. We also strengthen a result of V. Welker on the barycentric subdivision of collapsible complexes. We obtain this and other results on collapsibility of polyhedra by means of the characterization of the different notions of collapses in terms of finite topological spaces. © 2011 Springer Science+Business Media, LLC.

Registro:

Documento: Artículo
Título:Strong Homotopy Types, Nerves and Collapses
Autor:Barmak, J.A.; Minian, E.G.
Filiación:Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina
Palabras clave:Collapses; Finite spaces; Nerves; Non-evasiveness; Posets; Simple homotopy types; Simplicial actions; Simplicial complexes
Año:2012
Volumen:47
Número:2
Página de inicio:301
Página de fin:328
DOI: http://dx.doi.org/10.1007/s00454-011-9357-5
Título revista:Discrete and Computational Geometry
Título revista abreviado:Discrete Comput. Geom.
ISSN:01795376
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_01795376_v47_n2_p301_Barmak

Referencias:

  • Barmak, J.A., (2009) Algebraic topology of finite topological spaces and applications, , PhD thesis, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires
  • Barmak, J.A., Minian, E.G., Simple homotopy types and finite spaces (2008) Adv. Math., 218 (1), pp. 87-104
  • Barmak, J.A., Minian, E.G., One-point reductions of finite spaces, h-regular CW-complexes and collapsibility (2008) Algebr. Geom. Topol., 8, pp. 1763-1780
  • Björner, A., Topological methods (1995) Handbook of Combinatorics, 2, pp. 1819-1872
  • Cohen, M.M., (1970) A Course in Simple Homotopy Theory, , New York: Springer
  • Dowker, C.H., Homology groups of relations (1952) Ann. Math., 56, pp. 84-95
  • Escalante, F., Uber iterierte Cliquen-Graphen (1973) Abh. Math. Semin. Univ. Hamb., 39, pp. 58-68
  • Floyd, E.E., Richardson, R.W., An action of a finite group on an n-cell without stationary points (1959) Bull. Am. Math. Soc., 65, pp. 73-76
  • Frucht, R., Herstellung von Graphen mit vorgegebener abstrakter Gruppe (1939) Compos. Math., 6, pp. 239-250
  • Ginsburg, J., Dismantlability revisited for ordered sets and graphs and the fixed-clique property (1994) Can. Math. Bull., 37 (4), pp. 473-481
  • Glaser, L., (1970) Geometrical Combinatorial Topology, , Princeton: Van Nostrand Reinhold Company
  • Grünbaum, B., Nerves of simplicial complexes (1970) Aequ. Math., 4, pp. 63-73
  • Hachimori, M., (2000) Constructibility of constructible complexes, , PhD thesis, Graduate School of Arts and Sciences, the University of Tokyo
  • Hardie, K.A., Vermeulen, J.J.C., Homotopy theory of finite and locally finite T 0-spaces (1993) Expo. Math., 11, pp. 331-341
  • Kahn, J., Saks, M., Sturtevant, D., A topological approach to evasiveness (1984) Combinatorica, 4, pp. 297-306
  • Kozlov, D., (2007) Combinatorial Algebraic Topology, , New York: Springer
  • Lutz, F.H., Examples of ℤ-acyclic and contractible vertex-homogeneous simplicial complexes (2002) Discrete Comput. Geom., 27 (1), pp. 137-154
  • Lutz, F.H., Some results related to the evasiveness conjecture (2001) J. Comb. Theory, Ser. B, 81 (1), pp. 110-124
  • Matoušek, J., LC reductions yield isomorphic simplicial complexes (2008) Contrib. Discret. Math., 3 (2), pp. 37-39
  • May, J.P., Finite topological spaces, , http://www.math.uchicago.edu/~may/MISCMaster.html, Notes for REU (2003). Available at
  • May, J.P., Finite spaces and simplicial complexes, , http://www.math.uchicago.edu/~may/MISCMaster.html, Notes for REU (2003). Available at
  • McCord, M.C., Singular homology groups and homotopy groups of finite topological spaces (1966) Duke Math. J., 33, pp. 465-474
  • Oliver, R., Fixed-point sets of group actions on finite acyclic complexes (1975) Comment. Math. Helv., 50, pp. 155-177
  • Prisner, E., Convergence of iterated clique graphs (1992) Discrete Math., 103, pp. 199-207
  • Segev, Y., Some remarks on finite 1-acyclic and collapsible complexes (1994) J. Comb. Theory, Ser. A, 65, pp. 137-150
  • Spanier, E., (1966) Algebraic Topology, , Berlin: Springer
  • Stong, R.E., Finite topological spaces (1966) Trans. Am. Math. Soc., 123, pp. 325-340
  • Stong, R.E., Group actions on finite spaces (1984) Discrete Math., 49, pp. 95-100
  • Welker, V., Constructions preserving evasiveness and collapsibility (1999) Discrete Math., 207, pp. 243-255
  • Whitehead, J.H.C., Simplicial spaces, nuclei and m-groups (1939) Proc. Lond. Math. Soc., 45, pp. 243-327
  • Zeeman, E.C., On the dunce hat (1964) Topology, 2, pp. 341-358
  • Zeeman, E.C., Seminar on combinatorial topology (1963) Mimeographed notes, Inst. Hautes Études Sci., , Paris

Citas:

---------- APA ----------
Barmak, J.A. & Minian, E.G. (2012) . Strong Homotopy Types, Nerves and Collapses. Discrete and Computational Geometry, 47(2), 301-328.
http://dx.doi.org/10.1007/s00454-011-9357-5
---------- CHICAGO ----------
Barmak, J.A., Minian, E.G. "Strong Homotopy Types, Nerves and Collapses" . Discrete and Computational Geometry 47, no. 2 (2012) : 301-328.
http://dx.doi.org/10.1007/s00454-011-9357-5
---------- MLA ----------
Barmak, J.A., Minian, E.G. "Strong Homotopy Types, Nerves and Collapses" . Discrete and Computational Geometry, vol. 47, no. 2, 2012, pp. 301-328.
http://dx.doi.org/10.1007/s00454-011-9357-5
---------- VANCOUVER ----------
Barmak, J.A., Minian, E.G. Strong Homotopy Types, Nerves and Collapses. Discrete Comput. Geom. 2012;47(2):301-328.
http://dx.doi.org/10.1007/s00454-011-9357-5