# Vanishing homology of simplicial complexes with few facets

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 $0 \leq j \leq t-2\,$?

• Don't you have a 0-dim facet for each vertex (for a total of n)? What is the definition of facet? – Włodzimierz Holsztyński Sep 16 '14 at 8:25
• "Facets" are maximal faces. – Moty Katzman Sep 16 '14 at 9:39
• Thank you. (Sometimes facets may mean the next to the last dimension). – Włodzimierz Holsztyński Sep 16 '14 at 17:31
• Just in case (sorry): does maximal mean of maximal dimension or with respect to inclusion ? – 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 $n-t-2$