Let $K$ be a simplicial complex with $n$ vertices and $n-t$ facets, where $t \geq 3$.

Is it true that the $(n-3-j)$th reduced homology (with coefficients in a field) of $K$ vanishes for $0 \leq j \leq t-2\,$?

  • $\begingroup$ Don't you have a 0-dim facet for each vertex (for a total of n)? What is the definition of facet? $\endgroup$ – Włodzimierz Holsztyński Sep 16 '14 at 8:25
  • $\begingroup$ "Facets" are maximal faces. $\endgroup$ – Moty Katzman Sep 16 '14 at 9:39
  • $\begingroup$ Thank you. (Sometimes facets may mean the next to the last dimension). $\endgroup$ – Włodzimierz Holsztyński Sep 16 '14 at 17:31
  • $\begingroup$ Just in case (sorry): does maximal mean of maximal dimension or with respect to inclusion ? $\endgroup$ – Włodzimierz Holsztyński Sep 16 '14 at 17:45

By the nerve lemma, your complex is homotopy equivalent to a complex with $n-t$ vertices and therefore has trivial homology in degrees greater than $n-t-2$


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.