Topological rigidity of complex and quaternionic moment--angle manifolds

Abstract: We investigate the equivariant topological rigidity of complex and quaternionic moment--angle manifolds. By reducing the classification to the equivariant rigidity of their quasitoric (or quoric) quotients and the classification of the associated principal bundles, we establish new rigidity results within the category of locally linear actions. We prove that complex moment-angle manifolds are equivariantly rigid: any locally linear manifold equivariantly homotopy equivalent to a complex moment--angle manifold is equivariantly homeomorphic to it. In the quaternionic setting, we establish full equivariant rigidity for manifolds with four-dimensional quoric quotients and provide a primary rigidity statement for higher dimensions based on degree-4 characteristic classes. These results characterize moment--angle manifolds as equivariant strong Borel manifolds, demonstrating that their equivariant homotopy type completely determines their equivariant homeomorphism type.

• The paper proves that equivariant homotopy equivalences determine the equivariant homeomorphism type of complex moment–angle manifolds by combining quotient rigidity, characteristic-class classification, and bundle lifting.
• The paper establishes full quaternionic rigidity when the quoric quotient is 4-dimensional, while higher-dimensional cases retain only degree-4 primary invariants because principal $(S^3)^r$-bundles have additional obstructions.
• The paper shows that complex kernel bundles are completely determined by first Chern classes computed from the characteristic matrix, whereas finding analogous explicit quaternionic formulas remains an open problem.

This paper establishes equivariant topological rigidity results for complex and quaternionic moment–angle manifolds, showing that within the category of locally linear torus and quaternionic torus actions, the equivariant homotopy type of these spaces determines their equivariant homeomorphism type. The argument proceeds by a three-layer reduction: rigidity of the quasitoric or quoric quotient, classification of the principal kernel bundle over that quotient via characteristic classes, and lifting of the resulting homeomorphism to the total space.

Background and main statements

Moment–angle manifolds $\mathcal Z_P = (D^2,S^1)^K$ associated to a simple polytope $P$ with $m$ facets carry a natural $T^m$-action, and admit a free subtorus action of $K \cong (S^1)^{m-n}$ whose quotient is a quasitoric manifold over $P$. The quaternionic analogues $\mathcal Z_P^{\mathbb H} = (D^4,S^3)^K$, introduced by Hopkinson, carry locally regular actions of $Q^m=(S^3)^n$ and fiber over quoric manifolds via a free $(S^3)^{m-n}$-subgroup. The paper's central results are:

• Complex case: any locally linear $T^m$-manifold $P$0 admitting a $P$1-equivariant homotopy equivalence $P$2 is $P$3-equivariantly homeomorphic to $P$4; moreover $P$5 is $P$6-homotopic to an equivariant homeomorphism.
• Quaternionic case: full equivariant rigidity holds when the quoric quotient has dimension 4; in higher dimensions only a primary rigidity statement is obtained — equivariant homotopy equivalences preserve the degree-4 characteristic classes of the kernel bundle, so the corresponding principal $P$7-bundles agree at the level of primary invariants.

The asymmetry between the two settings stems from classifying-space topology: since $P$8, principal $P$9-bundles are completely classified by first Chern classes in $m$0, whereas $m$1 is not an Eilenberg–MacLane space, so degree-4 classes classify principal $m$2-bundles only over bases of dimension at most 4.

Reduction to quotients

The reduction layer rests on two structural facts proved for arbitrary locally linear $m$3-manifolds ($m$4 or $m$5) equipped with a $m$6-equivariant homotopy equivalence $m$7. First, an isotropy-transfer proposition shows that the action on $m$8 is effective, all nontrivial isotropy subgroups are coordinate subgroups of $m$9, and the isotropy data coincide with those of $T^m$0; this follows from Bredon's result that an equivariant homotopy equivalence induces homotopy equivalences on fixed-point sets. Second, if $T^m$1 acts freely on $T^m$2, then $T^m$3 acts freely on $T^m$4: any nontrivial subgroup $T^m$5 would give $T^m$6, contradicting $T^m$7 under the fixed-set homotopy equivalence. Consequently $T^m$8 descends to a homotopy equivalence $T^m$9 of quasitoric or quoric manifolds.

Applying the known rigidity theorems — Metaftsis–Prassidis for quasitoric manifolds and Gkeneralis–Prassidis for quoric manifolds — identifies the two quotients up to equivariant homeomorphism. The classification problem then reduces entirely to comparing principal kernel bundles over the common base.

Characteristic classes of kernel bundles

In the complex case, the characteristic matrix $K \cong (S^1)^{m-n}$0 defines a surjection $K \cong (S^1)^{m-n}$1 with kernel $K \cong (S^1)^{m-n}$2. For each facet $K \cong (S^1)^{m-n}$3 of $K \cong (S^1)^{m-n}$4, the characteristic submanifold $K \cong (S^1)^{m-n}$5 has codimension 2, and its Poincaré dual $K \cong (S^1)^{m-n}$6 equals the first Chern class of the associated line bundle $K \cong (S^1)^{m-n}$7. This is verified via the local Hopf model: a tubular neighborhood of $K \cong (S^1)^{m-n}$8 restricts the circle bundle to the Hopf fibration $K \cong (S^1)^{m-n}$9 over a transverse disc. The paper further shows, using Notbohm's line bundles over the Davis–Januszkiewicz space, that the facet bundles are pullbacks $P$0, so facet classes are pullbacks of Stanley–Reisner generators $P$1.

The key computation is the Chern class formula for the kernel bundle: choosing a matrix $P$2 whose rows generate $P$3,

$P$4

The proof uses Yoshida's framework of Euler classes for local torus actions, building transition functions from vertex charts of the polytope and applying the connecting homomorphism of the exponential sequence. Since principal $P$5-bundles are classified by $P$6, the kernel bundle is completely determined by $P$7 — a strong statement tying the bundle topology directly to combinatorial data. The relation to Yoshida's local Euler classes is clarified: because $P$8, the global Euler class vanishes, but the local facet contributions persist and coincide with the facet classes.

In the quaternionic case, the analogous construction yields quaternionic facet classes $P$9 for codimension-4 characteristic submanifolds, detected through the quaternionic Hopf fibration $\mathcal Z_P^{\mathbb H} = (D^4,S^3)^K$0 restricted to transverse 4-discs. The second Chern classes $\mathcal Z_P^{\mathbb H} = (D^4,S^3)^K$1 of the quaternionic line bundles associated to the factors of $\mathcal Z_P^{\mathbb H} = (D^4,S^3)^K$2 form the tuple of primary invariants in $\mathcal Z_P^{\mathbb H} = (D^4,S^3)^K$3. Crucially, the author concedes that no matrix formula analogous to the complex Chern formula is established here, leaving the explicit combinatorial description of these classes as an open problem. Over 4-dimensional bases, the tuple classifies the bundle completely (via the fact that $\mathcal Z_P^{\mathbb H} = (D^4,S^3)^K$4 is 3-connected with 4-type $\mathcal Z_P^{\mathbb H} = (D^4,S^3)^K$5); in higher dimensions it does not, as reflected in Granja's work on quaternionic line bundles and Crowley–Goette's secondary Kreck–Stolz invariants. The special case $\mathcal Z_P^{\mathbb H} = (D^4,S^3)^K$6 connects to the Feder–Gitler restrictions on degrees of self-maps of quaternionic projective spaces, which encode precisely the higher obstructions obstructing full classification.

Rigidity theorems

Combining the layers yields the complex rigidity theorem in full generality: after identifying the quasitoric quotients via the Metaftsis–Prassidis theorem, both $\mathcal Z_P^{\mathbb H} = (D^4,S^3)^K$7 and $\mathcal Z_P^{\mathbb H} = (D^4,S^3)^K$8 are principal $\mathcal Z_P^{\mathbb H} = (D^4,S^3)^K$9-bundles over the same base, and their first Chern classes agree because the base homeomorphism preserves characteristic submanifolds and hence facet classes. A corollary worth emphasizing: any locally linear $Q^m=(S^3)^n$0-manifold equivariantly homotopy equivalent to a complex moment–angle manifold is itself a complex moment–angle manifold — the class is closed under this equivalence relation.

The quaternionic rigidity theorem holds unconditionally when the quoric quotient has dimension 4, by the same argument with $Q^m=(S^3)^n$1 replacing $Q^m=(S^3)^n$2. In general dimension, the primary rigidity theorem asserts only that the induced quotient map is equivariantly homotopic to an equivariant homeomorphism and that the kernel bundles share their primary degree-4 classes; full rigidity would require controlling higher homotopy-theoretic invariants of maps into $Q^m=(S^3)^n$3.

Relation to the Borel property

The paper situates its results in the rigidity landscape of Kreck–Lück: although moment–angle manifolds are generally not aspherical and thus fall outside the classical Borel Conjecture, they satisfy the strong Borel property equivariantly — every equivariant homotopy equivalence from a locally linear manifold is homotopic to an equivariant homeomorphism. The author draws an analogy with the Farrell–Jones paradigm, suggesting that the "twisting" encoded in kernel-bundle characteristic classes constrains the topology analogously to how fundamental groups constrain aspherical manifolds. This analogy is suggestive rather than formalized, and the paper does not develop it into a general classification program.

Limitations and open questions

Several limitations are stated explicitly. The quaternionic rigidity is complete only for 4-dimensional quoric bases; the higher-dimensional statement is genuinely weaker, contingent on the unresolved classification of principal $Q^m=(S^3)^n$4-bundles over quoric manifolds beyond primary invariants. No explicit matrix formula for the quaternionic kernel classes is given, in contrast with the complex case. The distinction between the degree-4 kernel classes and the Euler class of the orbit map (in the sense of Batakidis–Gkeneralis) is noted but not fully exploited. The natural open question posed is whether principal $Q^m=(S^3)^n$5-bundles over quoric quotients of dimension greater than 4 admit a classification by explicit additional invariants; such a classification would be the missing ingredient for extending full quaternionic rigidity beyond dimension 4.

Conclusion

The paper demonstrates that complex moment–angle manifolds are equivariantly rigid among locally linear $Q^m=(S^3)^n$6-manifolds, and that quaternionic moment–angle manifolds are rigid when the quoric quotient is 4-dimensional, with primary-invariant rigidity in general. The method — reducing to quotient rigidity plus kernel-bundle classification — cleanly isolates exactly where the complex and quaternionic theories diverge, namely in the failure of $Q^m=(S^3)^n$7 to be an Eilenberg–MacLane space. The results establish moment–angle manifolds as equivariant strong Borel manifolds and reduce the remaining quaternionic gap to a concrete bundle-classification question.