Balanced complexes and complexes without large missing faces

TL;DR

This paper investigates face numbers of simplicial complexes with restrictions on missing faces, identifying minimal complexes and establishing bounds, with implications for Cohen--Macaulay and flag complexes.

Contribution

It characterizes minimal face vectors among complexes without large missing faces and provides bounds for 2-CM balanced complexes.

Findings

A polytopal sphere minimizes the f-vector among certain complexes.

The same sphere minimizes the h-vector among 2-CM complexes.

Lower bounds on face numbers for 2-CM balanced complexes are established.

Abstract

The face numbers of simplicial complexes without missing faces of dimension larger than $i$ are studied. It is shown that among all such $(d-1)$-dimensional complexes with non-vanishing top homology, a certain polytopal sphere has the componentwise minimal $f$-vector; and moreover, among all such 2-Cohen--Macaulay (2-CM) complexes, the same sphere has the componentwise minimal $h$-vector. It is also verified that the $l$-skeleton of a flag $(d-1)$-dimensional 2-CM complex is $2(d-l)$-CM while the $l$-skeleton of a flag PL $(d-1)$-sphere is $2(d-l)$-homotopy CM. In addition, tight lower bounds on the face numbers of 2-CM balanced complexes in terms of their dimension and the number of vertices are established.