Topological rigidity of complex and quaternionic moment--angle manifolds

TL;DR

This paper proves that complex and certain quaternionic moment--angle manifolds are completely determined by their equivariant homotopy type, establishing their strong topological rigidity.

Contribution

It introduces new rigidity results for complex and quaternionic moment--angle manifolds, linking their classification to quotient and bundle structures.

Findings

Complex moment-angle manifolds are equivariantly rigid.

Full equivariant rigidity established for quaternionic manifolds with 4D quotients.

Higher-dimensional rigidity relies on degree-4 characteristic classes.

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.