On modular homology of simplicial complexes: Saturation
This article is a continuation of [J. Comb. Theory, Ser. A 93, 350-370 (2001; Zbl 0971.06006)] where the authors considered a homology-like theory called modular homology applied to shellable simplicial complexes. There it was shown that the modular homology of a shellable complex is not determined by its \(h\)-vector. The article under review should be read together with that article. In the article under review, it is shown that the homology of a shellable complex can nonetheless be embedded in a module constructed purely from the shelling of the complex. From this, the authors are able to prove certain simple bounds on the Betti numbers of the complex. Complexes attaining these bounds are called saturated complexes. The authors also study conditions under which a shellable complex is saturated, and are able to give sufficient conditions under which gluing a simplex to a saturated complex yields still another saturated complex. Towards the end, the results are applied to Coxeter complexes and buildings.
scientific article; zbMATH DE number 1760687