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

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  • $\begingroup$ 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
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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 $n-t-2$

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