Equivariant rigidity of complex and quaternionic moment--angle manifolds

Abstract

We investigate equivariant rigidity properties of complex and quaternionic moment--angle manifolds. By reducing the classification problem to the equivariant rigidity of their quasitoric or quoric quotients and by using the associated principal bundle structures, we establish rigidity results within the category of locally linear actions. We prove that complex moment--angle manifolds are equivariantly rigid up to homeomorphism: any closed locally linear manifold equivariantly homotopy equivalent to a complex moment--angle manifold is equivariantly homeomorphic to it. In the quaternionic setting, under Hopkinson's globality condition on the characteristic data, we obtain the analogous equivariant homeomorphism rigidity for quaternionic moment--angle manifolds. These results show that, in the equivariant category considered here, the equivariant homotopy type of a moment--angle manifold determines its equivariant homeomorphism type.