Remarks on Topological Rigidity of Real Moment-Angle Manifolds

Abstract: We study topological rigidity of real moment-angle manifolds associated to flag simplicial complexes. Using the cubical geometry arising from the Davis construction, we identify the universal cover with the Davis complex and deduce that it admits a CAT(0) metric. As a consequence, its fundamental group satisfies the Farrell--Jones conjecture. Applying surgery theory, we deduce that real moment-angle manifolds of dimension at least five associated to flag complexes satisfy the Borel Conjecture. We also explain why this rigidity phenomenon is specific to the real case and fails for complex and quaternionic moment-angle complexes.

• The paper establishes that real moment-angle manifolds built from flag complexes are aspherical, uniquely determining their topology.
• It employs cubical CAT(0) geometry to link the manifold’s structure with right-angled Coxeter groups via the Davis complex.
• Rigidity is confirmed through surgery theory and the Farrell–Jones conjecture, ensuring any homotopy equivalence is homotopic to a homeomorphism in dimensions five and above.

Topological Rigidity of Real Moment-Angle Manifolds Associated with Flag Complexes

Introduction and Context

The paper investigates the topological rigidity of real moment-angle manifolds constructed from flag simplicial complexes. Moment-angle manifolds serve as central objects in toric topology, encoding combinatorial data from simplicial complexes or polytopes via polyhedral products. The real variant, associated with the pair $(D^1, S^0)$, reveals large-scale geometric properties distinct from its complex and quaternionic analogs when the underlying complex is flag.

For flag complexes, the real moment-angle manifold $\mathcal{R}_K = Z_K(D^1, S^0)$ admits asphericity, i.e., its universal cover is contractible. This asphericity enables the manifold to possess a CAT(0) geometric structure derived from the Davis complex of right-angled Coxeter groups, with profound implications for its fundamental group and subsequent topological rigidity.

Structural Properties and Asphericity

The paper rigorously establishes that the real moment-angle manifold $\mathcal{R}_K$ is aspherical if and only if $K$ is flag, leveraging Davis's criterion for polyhedral products. The combinatorial flag condition ensures that every pairwise-linked set of vertices in $K$ forms a simplex, which is necessary for the manifold to be aspherical.

This asphericity is not generic across all moment-angle constructions. In the complex $(D^2, S^1)$ and quaternionic $(D^4, S^3)$ cases, the inclusion maps do not satisfy injectivity or asphericity criteria in Davis's theorem—breakdown arises due to trivial or non-aspherical boundary homotopy groups. Consequently, neither complex nor quaternionic moment-angle manifolds inherit the rigidity characteristics accessible in the real context.

Cubical Geometry and CAT(0) Structure

Real moment-angle manifolds associated with flag complexes inherit a cubical structure, permitting a precise identification of their universal cover with the Davis complex of the corresponding right-angled Coxeter group $W_K$. Each vertex's link in the cubical complex is isomorphic to $K$, and Gromov's link condition ensures non-positive curvature; Moussong's theorem further guarantees a CAT(0) metric.

The fundamental group $\pi_1(\mathcal{R}_K)$, thus, acts properly, cocompactly, and isometrically on this finite-dimensional CAT(0) space. This geometric action is pivotal to the subsequent application of rigidity theorems.

Rigidity via Farrell–Jones and Surgery Theory

The geometric CAT(0) structure situates $\mathcal{R}_K = Z_K(D^1, S^0)$0 within classes of groups for which the Farrell–Jones conjecture in $\mathcal{R}_K = Z_K(D^1, S^0)$1-theory is known to hold. Bartels and Lück's results confirm this assertion for CAT(0) groups, establishing that the assembly map in $\mathcal{R}_K = Z_K(D^1, S^0)$2-theory is an isomorphism. This property is not merely algebraic: it has strong consequences for the topology of aspherical manifolds.

Applying surgery theory, the paper demonstrates that closed real moment-angle manifolds of dimension at least five, associated to flag complexes, have trivial structure sets. Thus, any homotopy equivalence between such manifolds is homotopic to a homeomorphism—formalizing topological rigidity in the non-equivariant Borel sense.

Notably, this approach depends solely on the structure and geometry of the fundamental group, without necessity for group actions on the target manifold. The rigidity result is distinct from classical equivariant rigidity in Coxeter-theoretic settings, providing a complementary viewpoint.

Implications and Further Developments

The results identify a natural class of aspherical manifolds, accessible by methods from geometric group theory and surgery theory, for which rigidity holds. This bridges toric topology with rigidity theory, and contrasts with the majority of spaces studied in toric topology (e.g., quasitoric manifolds), which are not aspherical and thus outside the scope of the Borel Conjecture.

The paper also highlights the specificity of rigidity to real moment-angle manifolds: complex and quaternionic cases fail to satisfy the requisite geometric or combinatorial conditions. This suggests a broader problem—determining which aspherical polyhedral products admit sufficient geometric structure for fundamental groups to satisfy Farrell–Jones and hence rigidity. The identification of additional classes remains open and worthy of further exploration.

Numerical Results and Strong Claims

• The paper proves that $\mathcal{R}_K = Z_K(D^1, S^0)$3 is aspherical if and only if $\mathcal{R}_K = Z_K(D^1, S^0)$4 is flag.
• The Farrell–Jones conjecture holds for fundamental groups of these manifolds.
• Any homotopy equivalence between such closed manifolds (dimension $\mathcal{R}_K = Z_K(D^1, S^0)$5) is homotopic to a homeomorphism.
• This rigidity is exclusive to the real case; analogous statements fail for complex/quaternionic moment-angle manifolds.

Conclusion

The paper achieves a rigorous proof of topological rigidity for real moment-angle manifolds associated to flag complexes, contingent on CAT(0) geometry and the Farrell–Jones conjecture. This contributes a distinguished subclass of toric-topological objects to rigidity theory, formalizes non-equivariant Borel rigidity, and underscores the geometric combinatorial foundations underlying large-scale manifold behavior. Future developments may extend these techniques to broader classes of aspherical polyhedral products, with implications for geometric group theory, manifold topology, and related domains.