Afshin Goodarzi: On face numbers of simplicial complexes satisfying Serre's conditions

Abstract

Cohen-Macaulayness and normality are very important properties of simplicial complexes. For instance every triangulation of a topological manifold is normal and Stanley's proof of Upper Bound Theorem is based on Cohen-Macaulayness of triangulated spheres. Both of these properties can be defined by vanishing certain homology groups of the link of faces. In fact, every Cohen-Macaulay complex is normal. But the gap between theses two is huge as for Cohen-Macaulayness one requires for each link all except possibly one homology group to vanish while normality means to have connected links. 

There is a spectrum of properties, Serre's conditions (S_r), filling the gap between Cohen-Macaulay property and normality. 

In this talk, I will introduce these properties. Then I will give some conditions for a given integer vector to be the f-vector of a complex with Serre's condition (S_r). This talk is based on a joint work with Pournaki, Fakhari and Yassemi.