$LS$-category of moment-angle manifolds and higher order Massey products

TL;DR

This paper investigates the LS category of moment-angle complexes using combinatorics and Massey products, providing bounds, characterizations, and applications to specific classes of manifolds.

Contribution

It offers a systematic combinatorial approach to determine the LS category of moment-angle complexes and explores the relationship with Massey products, extending to various classes of manifolds.

Findings

Bounds for LS category of moment-angle complexes

Characterization of LS category for complexes over triangulated manifolds and spheres

Conditions for vanishing of Massey products in specific examples

Abstract

Using the combinatorics of the underlying simplicial complex $K$, we give various upper and lower bounds for the Lusternik-Schnirelmann (LS) category of moment-angle complexes $\zk$. We describe families of simplicial complexes and combinatorial operations which allow for a systematic description of the LS category. In particular, we characterise the LS category of moment-angle complexes $\zk$ over triangulated $d$-manifolds $K$ for $d\leq 2$, as well as higher dimension spheres built up via connected sum, join, and vertex doubling operations. %This characterisation is given in terms of the combinatorics of $K$, the cup product length of $H^*(\zk)$, as well as a certain Massey products. We show that the LS category closely relates to vanishing of Massey products in $H^*(\zk)$ and through this connection we describe first structural properties of Massey products in moment-angel manifolds. Some of further applications include calculations of the LS category and the description of conditions for vanishing of Massey products for moment-angle manifolds over fullerenes, Pogorelov polytopes and $k$-neighbourly complexes, which double as important examples of hyperbolic manifolds.