Smoothability of moment-angle manifolds over sphere triangulations

For a finite simplicial complex $K$ on $m$ vertices, the moment-angle complex $\mathcal{Z}_K=\bigcup_{\sigma\in K}(D^2)^\sigma\times(S^1)^{[m]\setminus\sigma}$ is a central object of toric topology. When $K$ triangulates a $(d-1)$-sphere, or more generally is a generalized homology sphere, $\mathcal{Z}_K$ is a closed topological manifold of dimension $m+d$, and it is known to admit a smooth structure whenever $K$ is polytopal or is the underlying complex of a complete simplicial fan. Whether $\mathcal{Z}_K$ admits any smooth structure for the remaining sphere triangulations is a well-known open problem, posed as Question 9.3 in Panov's 2013 survey and restated as open in Fan's "Toric spaces and face enumeration on simplicial manifolds" (arXiv:2001.08390).

Background

Toric topology studies spaces with torus actions built from finite combinatorial data. Its central object is the moment-angle complex: for a simplicial complex $K$ on the vertex set $\{1,\dots,m\}$,

$\mathcal{Z}_K=\bigcup_{\sigma\in K}(D^2)^{\sigma}\times(S^1)^{[m]\setminus\sigma}\ \subset\ (D^2)^m,$

where a point of $(D^2)^m$ belongs to $\mathcal{Z}_K$ if for some simplex $\sigma$ of $K$ the coordinates indexed by $\sigma$ lie in the disc $D^2$ and the remaining coordinates lie on the boundary circle $S^1$. The torus $T^m=(S^1)^m$ acts naturally on $\mathcal{Z}_K$, and quotients of $\mathcal{Z}_K$ by freely acting subtori produce quasitoric manifolds, the topological analogues of toric varieties introduced by Davis and Januszkiewicz; thus $\mathcal{Z}_K$ plays the role of a homogeneous-coordinate and covering object for a large family of manifolds with torus actions.

A triangulated $(d-1)$-sphere is a simplicial complex whose geometric realization is homeomorphic to $S^{d-1}$; it is called polytopal if it is the boundary complex of a simplicial convex polytope. A generalized homology sphere is a simplicial complex that is a homology manifold (the link of every simplex has the same homology as a sphere of the complementary dimension) and has the homology of a sphere. For these complexes the moment-angle construction yields manifolds: if $K$ triangulates a $(d-1)$-sphere on $m$ vertices, then $\mathcal{Z}_K$ is a closed orientable topological manifold of dimension $m+d$ (Theorem 7.8 in Panov's survey), and Li Cai extended this manifold property to generalized homology spheres.

Smooth structures are known in two overlapping regimes. If $K$ is polytopal (the boundary complex of a simplicial polytope, equivalently of the polar of a simple polytope), then $\mathcal{Z}_K$ is realized as a nondegenerate intersection of Hermitian quadrics in $\mathbb{C}^m$, the level set of a moment map, and is a smooth $T^m$-equivariant manifold (Bosio and Meersseman; see Panov's survey). More generally, if $K$ is the underlying complex of a complete simplicial fan $\Sigma$ — a finite collection of simplicial cones covering $\mathbb{R}^d$ pairwise along common faces — then $\mathcal{Z}_K$ is smooth, realized as a quotient of an open subset of $\mathbb{C}^m$ by a real $(m-d)$-dimensional subgroup (Panov's survey, Theorem 9.2). Complete fans need not be projective, so the fan-realizable spheres form a class strictly larger than the polytopal spheres. Crucially, Panov also observes that this quotient construction produces a Hausdorff smooth manifold precisely for fan data, so for sphere triangulations supporting no complete fan no known mechanism yields any smooth structure at all.

For a closed topological manifold $M$ of dimension at least 5, smoothability is governed by the Kirby–Siebenmann class $\mathrm{ks}(M)\in H^4(M;\mathbb{Z}/2)$: $M$ admits a smooth structure if and only if $\mathrm{ks}(M)=0$ (dimension 4 is exceptional and requires different methods). No technique is known for computing Kirby–Siebenmann classes of moment-angle manifolds from the combinatorics of $K$, and Hasui, Kishimoto and Kizu proved that all Stiefel–Whitney classes of moment-angle manifolds are trivial — these are the standard mod-2 characteristic classes of a manifold, the simplest invariants sensitive to smoothing — so this family of invariants carries no information here.

The question asked below arises from Feifei Fan's study of toric spaces and face enumeration (arXiv:2001.08390), which constructs "rational toric manifolds" — quotient spaces of $\mathcal{Z}_\Delta$ generalizing quasitoric manifolds to homology spheres — as a topological model for the algebra of face rings, and which states explicitly that beyond the polytopal case "the smoothness of $\mathcal{Z}_\Delta$ is open". The formulation given here is the standard one in the field (Panov's Question 9.3).

Problem Statement

Let $K$ be a triangulation of a $(d-1)$-sphere on $m$ vertices — more generally, a generalized homology sphere — so that the moment-angle complex $\mathcal{Z}_K$ is a closed topological manifold of dimension $m+d$. It is known that $\mathcal{Z}_K$ admits a smooth structure whenever $K$ is polytopal or is the underlying complex of a complete simplicial fan.

(a) Does $\mathcal{Z}_K$ admit a smooth structure — not required to be $T^m$-equivariant and not required to arise from the quotient construction — for every sphere triangulation $K$, in particular for non-polytopal spheres and for homology spheres that support no complete fan?

(b) More generally, as posed by Panov (Question 9.3): describe the class of sphere triangulations $K$ for which the moment-angle manifold $\mathcal{Z}_K$ admits a smooth structure.

The question is resolved by a construction of a smooth structure on $\mathcal{Z}_K$ for arbitrary sphere triangulations, by a sphere triangulation together with a proof that the topological manifold $\mathcal{Z}_K$ is not smoothable, or by necessary and sufficient conditions on $K$ that characterize smoothability.

References

1. Feifei Fan, "Toric spaces and face enumeration on simplicial manifolds", arXiv:2001.08390 (2020). https://arxiv.org/abs/2001.08390
2. Taras E. Panov, "Geometric structures on moment-angle manifolds", Russian Mathematical Surveys 68 (2013), no. 3, 503–568, doi:10.1070/rm2013v068n03abeh004840; arXiv:1302.2463. https://arxiv.org/abs/1302.2463
3. Frédéric Bosio and Laurent Meersseman, "Real quadrics in C^n, complex manifolds and convex polytopes", Acta Mathematica 197 (2006), no. 1, 53–127, doi:10.1007/s11511-006-0008-2. https://doi.org/10.1007/s11511-006-0008-2
4. Li Cai, "On products in a real moment-angle manifold", Journal of the Mathematical Society of Japan 69 (2017), no. 2, 503–528, doi:10.2969/jmsj/06920503; arXiv:1410.5543. https://arxiv.org/abs/1410.5543
5. Michael W. Davis and Tadeusz Januszkiewicz, "Convex polytopes, Coxeter orbifolds and torus actions", Duke Mathematical Journal 62 (1991), no. 2, 417–451, doi:10.1215/s0012-7094-91-06217-4. https://doi.org/10.1215/s0012-7094-91-06217-4
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9. Stavros Argyrios Papadakis and Vasiliki Petrotou, "The characteristic 2 anisotropicity of simplicial spheres", arXiv:2012.09815 (2020). https://arxiv.org/abs/2012.09815
10. Kalle Karu and Elizabeth Xiao, "On the anisotropy theorem of Papadakis and Petrotou", Algebraic Combinatorics 6 (2023), no. 5, 1313–1330, doi:10.5802/alco.298; arXiv:2204.07758. https://arxiv.org/abs/2204.07758
11. Sho Hasui, Daisuke Kishimoto, and Akatsuki Kizu, "The Stiefel–Whitney classes of moment-angle manifolds are trivial", Forum Mathematicum 34 (2022), no. 6, 1463–1474, doi:10.1515/forum-2021-0267; arXiv:2110.06358. https://arxiv.org/abs/2110.06358
12. Nikolai Erokhovets and Elena Erokhovets, "On uniqueness of the equivariant smooth structure on a real moment-angle manifold", arXiv:2503.00446 (2025). https://arxiv.org/abs/2503.00446
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