Classification of moment-angle manifolds that are connected sums of sphere products

Characterize the class of simple polytopes and, more generally, simplicial spheres for which the associated moment-angle manifold is diffeomorphic, homeomorphic, or homotopy equivalent to a connected sum of products of spheres.

Background

The paper studies when a moment-angle manifold associated with a simplicial sphere or the nerve of a simple polytope has the topology of a connected sum of products of spheres. Although several substantial families are known—including polygons, dual-stacked polytopes, certain neighbourly polytopes, and the families established in the paper—the general classification remains unresolved.

The problem concerns several levels of equivalence: diffeomorphism in the smooth category, homeomorphism in the topological category, and homotopy equivalence in the weakest formulation. The paper contributes new diffeomorphism results for starshaped spheres satisfying specific combinatorial conditions, but does not provide a complete classification.

References

Nevertheless, it remains an open problem to describe the class of simple polytopes $P$ (or, more generally, simplicial spheres~$K$) for which the moment-angle manifold $Z_P$ is diffeomorphic (or, in a weaker formulation, homeomorphic or homotopy equivalent) to a connected sum of products of spheres.

— New families of moment-angle manifolds diffeomorphic to connected sums of products of spheres  (2609.19945 - Kovyrshina et al., 17 Sep 2026) in Section 1, Introduction