Moment-angle manifolds corresponding to three-dimensional simplicial spheres, chordality and connected sums of products of spheres

TL;DR

This paper characterizes when the cohomology ring of a moment-angle complex from a 3D simplicial sphere matches that of a connected sum of products of spheres, based on combinatorial properties of the sphere.

Contribution

It provides a complete characterization for 3D simplicial spheres and a sufficient condition for higher dimensions relating cohomology rings to connected sums of spheres.

Findings

Cohomology ring isomorphism occurs only for specific combinatorial conditions.

Characterization includes boundary of 4D cross-polytope, chordal graphs, and specific missing edges.

Sufficient conditions are given for higher-dimensional simplicial spheres.

Abstract

We prove that the moment-angle complex $\mathcal Z_K$ corresponding to a 3-dimensional simplicial sphere $K$ has the cohomology ring isomorphic to the cohomology ring of a connected sum of products of spheres if and only if either (a) $K$ is the boundary of a 4-dimensional cross-polytope, or (b) the one-skeleton of $K$ is a chordal graph, or (c) there are only two missing edges in $K$ and they form a chordless 4-cycle. For simplicial spheres $K$ of arbitrary dimension, we obtain a sufficient condition for the ring isomorphism $H^*(\mathcal Z_K)\cong H^*(M)$ where $M$ is a connected sum of products of spheres.