I would like to know if there are nice criteria to know if the ordered complex $C$ induced by a poset is contractible. I am also interested in the same question for subcomplexes of $C$.

$C$ happens to be a flag complex, and I found that all induced subcomplexes are contractible iff the underlying graph of $C$ is chordal.

However, this doesn't fit my purposes.

Do you know any result in this respect?

  • 5
    $\begingroup$ Glib answer: it's contractible if the poset has a minimal or maximal element :) $\endgroup$ Feb 27, 2019 at 16:22
  • 4
    $\begingroup$ There are lots of results that are useful, like Quillen's theorem A. Bjorner's handbook chapter on poset topology is also quite useful. Note that for a flag complex, chordal is equivalent to all INDUCED subcomplexes being contractible, not arbitrary ones. $\endgroup$ Feb 27, 2019 at 16:48
  • 2
    $\begingroup$ Actually, a simplicial complex must be a flag complex for all induced subcomplexes to be contractible. Otherwise, you have a clique in the 1-skeleton which is not the boundary of a simplex and that induced subgraph is not contractible. $\endgroup$ Feb 27, 2019 at 16:53
  • $\begingroup$ whoops, above I of course mean "minimum or maximum" (or "unique minimal or unique maximal element") $\endgroup$ Mar 1, 2019 at 20:24
  • $\begingroup$ Baclawski and Bjorner, "Fixed Points in Partially Ordered Sets," Advances in Mathematics 31 (1979) contains related results on page 271. $\endgroup$
    – Tri
    May 14, 2020 at 21:40


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