Pontryagin Algebras and the LS-Category of Moment–Angle Complexes in the Flag Case

For any flag simplicial complex $\mathcal K$, we describe the multigraded Poincaré series, the minimal number of relations, and the degrees of these relations in the Pontryagin algebra of the corresponding moment–angle complex $\mathcal Z_{\mathcal K}$. We compute the LS-category of $\mathcal Z_{\mathcal K}$ 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 page for flag $\mathcal K$. 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 $(\mathbb Z\times \mathbb Z_{\geq 0}^m)$-grading on the Pontryagin algebras which is similar to the multigrading on the cohomology of $\mathcal Z_{\mathcal K}$.