Separation-type combinatorial invariants for triangulations of manifolds

Giulia Codenotti, Francisco Santos and Jonathan Spreer

Abstract

We propose and study a variation of Bagchi and Datta's σ-vector of a simplicial complex C, whose entries are defined as weighted sums of Betti numbers of induced subcomplexes of C. We prove that these invariants satisfy an Alexander-Dehn-Sommerville type identity, and behave nicely under natural operations on triangulated manifolds and spheres such as connected sums and bistellar flips. In the language of commutative algebra, the invariants are weighted sums of graded Betti numbers of the Stanley-Reisner ring of C. This interpretation implies, by a result of Migliore and Nagel, that the Billera-Lee sphere maximizes these invariants among polytopal spheres with a given f-vector. We provide theoretical and experimental evidence that this bound also holds for non-polytopal spheres, and establish a slightly weaker upper bound for arbitrary pure and strongly connected simplicial complexes. Concerning lower bonds, our experimental evidence shows that these depend on whether only polytopal or arbitrary triangulated spheres are considered. As an application of the upper bound, we show how this can be used to obtain lower bounds on the f-vector of a triangulated 4-manifold with transitive symmetry and prescribed vector of Betti numbers.

AMS Subject Classification: Primary 57Q15; secondary 05E45, 13F55, 57M15.