Bosio–Meersseman conjecture for neighbourly polytopal spheres

Determine whether, for every neighbourly polytopal sphere $K_P$, the associated moment-angle manifold $Z_P$ is a connected sum of products of spheres, including cases not covered by the known results for polytopal spheres of odd dimension.

Background

The paper recalls a conjecture formulated by Bosio and Meersseman asserting that the moment-angle manifold of a neighbourly polytopal sphere is a connected sum of products of spheres. The conjecture was known in the case of polytopal spheres of odd dimension, where it had already been proved by Gitler and Lopez de Medrano.

The paper proves a smooth diffeomorphism result for odd-dimensional neighbourly starshaped spheres, thereby extending the known odd-dimensional picture beyond the polytopal setting. It does not resolve the conjecture in the remaining dimensions or for all neighbourly polytopal spheres. A later remark also notes that torsion in the relevant middle-dimensional homology could yield a counterexample, underscoring the unresolved status of the general claim.