Online Talk

In this talk we investigate rigidity phenomena for real, complex and quaternionic moment-angle manifolds from both equivariant and non-equivariant perspectives. 

For complex and quaternionic moment-angle manifolds, we study equivariant topological rigidity under locally linear torus actions. By reducing the classification problem to the equivariant rigidity of the associated quasitoric (respectively, quoric) quotients together with the classification of the corresponding principal bundles, we establish new rigidity results. In particular, we prove that complex moment-angle manifolds are equivariantly rigid: any locally linear manifold that is equivariantly homotopy equivalent to such a manifold is in fact equivariantly homeomorphic to it. In the quaternionic setting, we obtain full equivariant rigidity in dimension four (at the level of quoric quotients), and a primary rigidity result in higher dimensions governed by degree-4 characteristic classes. As a consequence, these manifolds are equivariant strong Borel manifolds, meaning that their equivariant homotopy type determines their equivariant homeomorphism type. 

In contrast, for real moment-angle manifolds associated to flag simplicial complexes, we establish (non-equivariant) topological rigidity. Using the cubical geometry arising from the Davis construction, we identify the universal cover with the Davis complex and show that it admits a CAT(0) metric. This implies that the fundamental group satisfies the Farrell-Jones conjecture. By applying surgery theory, we deduce that real moment-angle manifolds of dimension at least five associated to flag complexes satisfy the Borel Conjecture. 

Finally, we explain why this rigidity phenomenon is specific to the real case and does not extend to the complex and quaternionic settings.