×

Comparing minimal simplicial models. (English) Zbl 1279.55006

For a given topological space \(X\), this paper studies relations between the minimal sizes (in number of vertices) of various types of simplicial complexes which have the same homotopy type as \(X\) : \(m_s(X)\) denotes the minimal number of vertices obtained when all simplicial complexes (having the homotopy type of \(X\)) are considered while \(m_f(X)\) (resp. \(m_p(X)\)) denotes the minimum obtained when one restricts to flag complexes (resp. to partially ordered sets whose order complex is homotopy equivalent to \(X\)). The following inequalities follow from the definition of these invariants: \(m_s(X) \leq m_f(X)\leq m_p(X)\leq 2^{m_s(X)}\).
In this paper, among other results, the author proves (Theorem 1.2) that for any non-acyclic space \(X\) with \(m_s(X)\) finite : \[ \lim_{k \to +\infty} \frac{m_s(\Sigma^k X)}{k}=1~~~~\text{and}~~~~ \lim_{k \to +\infty} \frac{m_f(\Sigma^k X)}{k}=\lim_{k \to +\infty} \frac{m_p(\Sigma^k X)}{k}=2 \] (where \(\Sigma\) denotes the suspension). Actually, all these invariants are related to the homological dimension \(h(X)\) of \(X\), i.e. the maximal integer \(k\) such that \(\widetilde{H}_k(X,\Lambda)\) is non trivial for some group \(\Lambda\) (which is well defined for any non-acyclic space \(X\)). The author obtains certain characterizations concerning \(m_f(X)-2h(X)\) and \(m_s(X)-h(X)\) and the proofs give also interesting results about homology groups of flag complexes and their dependance on the number of vertices of the complex.

MSC:

55P10 Homotopy equivalences in algebraic topology
55U10 Simplicial sets and complexes in algebraic topology
05E45 Combinatorial aspects of simplicial complexes
55P40 Suspensions
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adamaszek, M.: Maximal Betti Number of a Flag Simplicial Complex. arxiv/1109.4775
[2] Barmak, J.A.: Algebraic Topology of Finite Topological Spaces and Applications, Lecture Notes in Mathematics, vol. 2032. Springer, Berlin-Heidelberg (2011) · Zbl 1235.55001
[3] Barmak J.A., Minian E.G.: Minimal finite models. J. Homotopy Relat. Struct 2(1), 127-140 (2007) · Zbl 1185.55005
[4] Hardie K.A., Vermeulen J.J.C., Witbooi P.J.: A nontrivial pairing of finite T0 spaces. Topol. Appl. 125(3), 533-542 (2002) · Zbl 1018.55007 · doi:10.1016/S0166-8641(01)00298-X
[5] Joswig M., Lutz F.H.: One-point suspensions and wreath products of polytopes and spheres. J. Combin. Theory Ser. A 110(2), 193-216 (2005) · Zbl 1095.57019 · doi:10.1016/j.jcta.2004.09.009
[6] Kahle M.: Topology of random clique complexes. Discrete Math. 309(6), 1658-1671 (2009) · Zbl 1215.05163 · doi:10.1016/j.disc.2008.02.037
[7] Katzman M.: Characteristic-independence of Betti numbers of graph ideals. J. Combin. Theory Ser. A 113, 435-454 (2006) · Zbl 1102.13024 · doi:10.1016/j.jcta.2005.04.005
[8] Kozlov D.: Convex hulls of f- and β-vector. Discrete Comput. Geom. 18(4), 421-431 (1997) · Zbl 0899.52010 · doi:10.1007/PL00009326
[9] Kozlov, D.: Combinatorial Algebraic Topology Algorithms and Computation in Mathematics, vol. 21. Springer, Berlin, Heidelberg (2008) · Zbl 1130.55001
[10] May, J.P.: Lecture notes about finite spaces for REU (2003). http://math.uchicago.edu/ may/finite.html
[11] McDuff D.: On the classifying spaces of discrete monoids. Topology 18(4), 313-320 (1979) · Zbl 0429.55009 · doi:10.1016/0040-9383(79)90022-3
[12] Stong R.E.: Finite topological spaces. Trans. Am. Math. Soc 123, 325-340 (1966) · Zbl 0151.29502 · doi:10.1090/S0002-9947-1966-0195042-2
[13] Weng, D.: On minimal finite models, a REU paper. http://www.math.uchicago.edu/ may/VIGRE/VIGRE2010/REUPapers/Weng.pdf
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.