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Small Flag Complexes with Torsion

Published online by Cambridge University Press:  20 November 2018

Michał Adamaszek
Michał Adamaszek*
Affiliation:
Fachbereich Mathematik, Universität Bremen, Bibliothekstr. 1, 28359 Bremen, Germany e-mail: [email protected]
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Abstract

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We classify flag complexes on at most 12 vertices with torsion in the first homology group. The result is moderately computer-aided.

As a consequence we confirm a folklore conjecture that the smallest poset whose order complex is homotopy equivalent to the real projective plane (and also the smallest poset with torsion in the first homology group) has exactly 13 elements.

Keywords

55U1006A1155P4055-0405-04clique complexorder complexhomologytorsionminimal model
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 57 , Issue 2 , 14 June 2014 , pp. 225 - 230
Copyright
Copyright © Canadian Mathematical Society 2014

Footnotes

This research was carried out when the author was a member of the Centre for Discrete Mathematics and its Applications (DIMAP) and the Mathematics Institute of the University of Warwick, Coventry, UK. The support of EPSRC award EP/D063191/1 is gratefully acknowledged.

References

[1] Barmak, J. A., Star clusters in independence complexes of graphs. Adv. Math. 241 (2013), 33–57. http://dx.doi.org/10.1016/j.aim.2013.03.016 CrossRefGoogle Scholar
[2] Barmak, J. A., Algebraic topology of finite topological spaces and applications. Lecture Notes in Mathematics, 2032, Springer-Verlag, Heidelberg, 2011.Google Scholar
[3] Barmak, J. A. and Minian, E. G., Minimal finite models. J. Homotopy Relat. Struct. 2 (2007), no. 1, 127–140.Google Scholar
[4] Computational Homology Project, http://chomp.rutgers.edu/ Google Scholar
[5] Engström, A., Independence complexes of claw-free graphs. European J. Combin. 29, no. 1, 234–241. http://dx.doi.org/10.1016/j.ejc.2006.09.007 Google Scholar
[6] Gawrilow, E. and Joswig, M., polymake: a framework for analyzing convex polytopes. In: Polytopes—combinatorics and computation (Oberwolfach, 1997), DMV Sem., 29, Birkhäuser, Basel, 2000, pp. 43–73.Google Scholar
[7] Hardie, K. A., Vermeulen, J. J. C., and Witbooi, P. J., A nontrivial pairing of finite T0 spaces. Topology Appl. 125 (2002), no. 3, 533–542. http://dx.doi.org/10.1016/S0166-8641(01)00298-X CrossRefGoogle Scholar
[8] Katzman, M., Characteristic-independence of Betti numbers of graph ideals. J. Combin. Theory Ser. A 113 (2006), no. 3, 435–454. http://dx.doi.org/10.1016/j.jcta.2005.04.005 CrossRefGoogle Scholar
[9] May, J. P., Lecture notes about finite spaces for REU. 2003, http://math.uchicago.edu/_may/finite.html Google Scholar
[10] McKay, B. D., The Nauty graph automorphism package. http://cs.anu.edu.au/_bdm/nauty/ Google Scholar
[11] The On-Line Encyclopedia of Integer Sequences, http://oeis.org Google Scholar
[12] Weng, D., On minimal finite models. a REU paper. http://www.math.uchicago.edu/_may/VIGRE/VIGRE2010/REUPapers/Weng.pdf Google Scholar