Average Betti numbers of induced subcomplexes in triangulations of manifolds

TL;DR

This paper introduces a weighted average of Betti numbers for induced subcomplexes in triangulated manifolds, revealing new algebraic identities and bounds that connect topological invariants with combinatorial and algebraic properties.

Contribution

It defines a new invariant related to Betti numbers, proves its key identities, and applies it to derive bounds on face numbers of triangulated manifolds with symmetry.

Findings

The invariants satisfy an Alexander-Dehn-Sommerville type identity.

Billera-Lee spheres maximize these invariants among triangulated spheres with fixed f-vector.

Upper bounds on the invariants lead to lower bounds on f-vectors of symmetric 4-manifolds.

Abstract

We study a variation of Bagchi and Datta's $\sigma$-vector of a simplicial complex $C$, whose entries are defined as weighted averages of Betti numbers of induced subcomplexes of $C$. We show 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 Adiprasito, that the Billera-Lee sphere maximizes these invariants among triangulated spheres with a given $f$-vector. For the first entry of $\sigma$, we extend this bound to the class of strongly connected pure complexes. As an application, we show how upper bounds on $\sigma$ can be used to obtain lower bounds on the $f$-vector of triangulated $4$-manifolds with transitive symmetry on vertices and prescribed vector of Betti numbers.