Complex moment-angle manifolds are equivariantly rigid, so their equivariant homotopy type determines their equivariant homeomorphism type, with similar but partial results for quaternionic moment-angle manifolds.
(answers to)Classification of SU(2)principal fibre bundles over four-dimensional manifolds, Mathoverflow URL (version: 2015-02-04):https://mathoverflow.net/q/195592
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Topological rigidity of complex and quaternionic moment--angle manifolds
Complex moment-angle manifolds are equivariantly rigid, so their equivariant homotopy type determines their equivariant homeomorphism type, with similar but partial results for quaternionic moment-angle manifolds.