Homology groups of simplicial complements (Q335033)

Let \(K\) be a simplicial complex on the set \([m]\). A missing face of \(K\) is the subset \(\tau\subseteq [m]\), where \(\tau\notin K\) and every proper subset of \(\tau\) is a simplex of \(K\). A simplicial complement \(P\) is a subset of all non-faces of \(K\) containing all missing faces. In the paper under review, the authors study the homology groups induced by the exterior algebra generated by the simplicial compliment of a simplicial complex \(K\). By using \(\text{C}\)ech homology and Alexander duality, it is shown that there is a duality between these homology groups and the simplicial homology groups of \(K\).