Face numbers of manifolds with boundary

TL;DR

This paper establishes sharp lower bounds on face numbers of simplicial complexes that triangulate manifolds with boundary, extending known results and introducing new invariants for relative complexes.

Contribution

It introduces new bounds on interior face numbers of manifolds with boundary and develops a version of $\sigma$- and $\mu$-numbers for relative complexes.

Findings

Sharp lower bounds on interior edges in terms of interior vertices and Betti numbers.

Extension of bounds to higher-dimensional interior faces for complexes with the weak Lefschetz property.

Development of $\sigma$- and $\mu$-numbers for relative simplicial complexes.

Abstract

We study face numbers of simplicial complexes that triangulate manifolds (or even normal pseudomanifolds) with boundary. Specifically, we establish a sharp lower bound on the number of interior edges of a simplicial normal pseudomanifold with boundary in terms of the number of interior vertices and relative Betti numbers. Moreover, for triangulations of manifolds with boundary all of whose vertex links have the weak Lefschetz property, we extend this result to sharp lower bounds on the number of higher-dimensional interior faces. Along the way we develop a version of Bagchi and Datta's $\sigma$- and $\mu$-numbers for the case of relative simplicial complexes and prove stronger versions of the above statements with the Betti numbers replaced by the $\mu$-numbers. Our results provide natural generalizations of known theorems and conjectures for closed manifolds and appear to be new even for the case of a ball.