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.
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Real moment-angle manifolds associated to flag complexes satisfy the Borel Conjecture in dimensions at least 5 because their universal covers admit CAT(0) metrics.
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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.
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Remarks on Topological Rigidity of Real Moment-Angle Manifolds
Real moment-angle manifolds associated to flag complexes satisfy the Borel Conjecture in dimensions at least 5 because their universal covers admit CAT(0) metrics.