Flag complexes and homology

TL;DR

This paper explores the relationships between face numbers and homology in flag complexes, establishing bounds and equivalences that extend previous theorems and provide new insights into their combinatorial and topological properties.

Contribution

It proves the existence of balanced complexes with matching face vectors and higher Betti numbers, and establishes bounds on Betti numbers and face vectors in flag complexes.

Findings

Existence of balanced complexes with same f-vector and higher Betti number.

Upper bounds on top-dimensional Betti numbers based on face numbers.

A continuous analog relating f-polynomial coefficients to Betti numbers.

Abstract

We prove several relations on the $f$-vectors and Betti numbers of flag complexes. For every flag complex $\Delta$, we show that there exists a balanced complex with the same $f$-vector as $\Delta$, and whose top-dimensional Betti number is at least that of $\Delta$, thereby extending a theorem of Frohmader by additionally taking homology into consideration. We obtain upper bounds on the top-dimensional Betti number of $\Delta$ in terms of its face numbers. We also give a quantitative refinement of a theorem of Meshulam by establishing lower bounds on the $f$-vector of $\Delta$, in terms of the top-dimensional Betti number of $\Delta$. This result has a continuous analog: If $\Delta$ is a $(d-1)$-dimensional flag complex whose $(d-1)$-th reduced homology group has dimension $a\geq 0$ (over some field), then the $f$-polynomial of $\Delta$ satisfies the coefficient-wise inequality $f_{\Delta}(x) \geq (1 + (\sqrt[d]{a}+1)x)^d$.