7
$\begingroup$

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?

$\endgroup$
5
  • 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

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.