Pontryagin algebras and the LS-category of moment-angle complexes in the flag case

TL;DR

This paper analyzes the algebraic and topological properties of moment-angle complexes associated with flag simplicial complexes, providing explicit descriptions of their Pontryagin algebras and LS-category, and extending previous results to arbitrary coefficient rings.

Contribution

It offers a detailed description of the Pontryagin algebra structure and LS-category for flag moment-angle complexes, and generalizes existing results to broader coefficient rings.

Findings

Pontryagin algebra structure explicitly described for flag complexes

LS-category computed for flag complexes and bounded in general case

Milnor-Moore spectral sequence collapses at second sheet for flag complexes

Abstract

For any flag simplicial complex $K$, we describe the multigraded Poincare series, the minimal number of relations and the degrees of these relations in the Pontryagin algebra of the corresponding moment-angle complex $Z_K$. We compute the LS-category of $Z_K$ for flag complexes and give a lower bound in the general case. The key observation is that the Milnor-Moore spectral sequence collapses at the second sheet for flag $K$. We also show that the results of Panov and Ray about the Pontryagin algebras of Davis-Januszkiewicz spaces are valid for arbitrary coefficient rings, and introduce the $(\mathbb{Z}\times\mathbb{Z}^m)$-grading on the Pontryagin algebras which is similar to the multigrading on the cohomology of $Z_K$.