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

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.