Skip to main content Accessibility help
×
Home
Hostname: page-component-55597f9d44-n4bck Total loading time: 0.23 Render date: 2022-08-12T04:08:23.066Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

Small Flag Complexes with Torsion

Published online by Cambridge University Press:  20 November 2018

Michał Adamaszek*
Affiliation:
Fachbereich Mathematik, Universität Bremen, Bibliothekstr. 1, 28359 Bremen, Germany e-mail: aszek@mimuw.edu.pl
Rights & Permissions[Opens in a new window]

Abstract

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Type
Research Article
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), 3357. 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, 127140.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, 234241. 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. 4373.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, 533542. 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, 435454. 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
You have Access
2
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Small Flag Complexes with Torsion
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Small Flag Complexes with Torsion
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Small Flag Complexes with Torsion
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *