# Criteria for a poset complex to be contractible

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?

• Glib answer: it's contractible if the poset has a minimal or maximal element :) – Sam Hopkins Feb 27 '19 at 16:22
• 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. – Benjamin Steinberg Feb 27 '19 at 16:48
• 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. – Benjamin Steinberg Feb 27 '19 at 16:53
• whoops, above I of course mean "minimum or maximum" (or "unique minimal or unique maximal element") – Sam Hopkins Mar 1 '19 at 20:24
• Baclawski and Bjorner, "Fixed Points in Partially Ordered Sets," Advances in Mathematics 31 (1979) contains related results on page 271. – Tri May 14 at 21:40