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

F. E. Vylegzhanin

Informatics and Automation, Toric Topology, Group Actions, Geometry, and Combinatorics. Part 1, Tome 317 (2022), pp. 64-88.

Résumé

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 $(\mathbb{Z}\times {\mathbb{Z}}_{\ge 0}^m)$-grading on the Pontryagin algebras which is similar to the multigrading on the cohomology of ${\mathcal{Z}}_{\mathcal{K}}$.

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