On face numbers of flag simplicial complexes

TL;DR

This paper revisits previous results on the homotopy groups and Poincaré series of moment-angle complexes associated with flag simplicial complexes, interpreting them as bounds on face numbers of such complexes.

Contribution

It provides a new interpretation of existing homotopy and Poincaré results as polynomial bounds on face numbers of flag simplicial complexes.

Findings

Homotopy group ranks and Poincaré series can be bounded polynomially

Reinterpretation of known results as face number bounds

Insights into face number limitations for flag complexes

Abstract

Denham, Suciu and Panov, Ray computed ranks of homotopy groups and Poincar\'e series of a moment-angle-complex $\mathcal Z(\mathcal K$) / Davis-Januzskiewicz space $DJ(\mathcal K)$ associated to a flag simplicial complex $\mathcal K$. In this note we revisit these results and interpret them as polynomial bounds on the face numbers of an arbitrary simplicial flag complex.