Separation-type combinatorial invariants for triangulations of manifolds

In my talk I will propose and discuss a set of combinatorial invariants of simplicial complexes. The invariants are very elementary and defined by counting connected components and/or homological features of induced subcomplexes, but admit a commutative algebra interpretation as weighted sums of graded Betti numbers of the underlying complex.

I will first define these invariants, state an Alexander-Dehn-Sommerville type identity they satisfy, and connect them to natural operations on triangulated manifolds and spheres. Then I will present a (non-optimal) upper bound for arbitrary pure and strongly connected simplicial complexes and discuss the very natural conjecture that, for triangulated spheres of a given f-vector, the invariants are maximised for the Billera-Lee-spheres.

This is joint work with Giulia Codenotti and Francisco Santos, see arxiv.org/abs/1808.04220.