Topological rigidity of quoric manifolds
I. Gkeneralis, S. Prassidis
TL;DR
The paper proves an equivariant rigidity theorem for quoric manifolds, quaternionic analogues of toric spaces, by developing a canonical-model framework and using the Euler class obstruction of the orbit map. The main result shows that a locally linear $Q^n$-manifold equivariantly homotopy equivalent to a quoric manifold is actually equivariantly homeomorphic to it, provided the base is a nice homotopy polytope. The approach extends methods from Coxeter and toric geometry to the quaternionic setting, leveraging regular corners, characteristic functors, and face-preserving maps, and culminates in a mechanism to lift equivariant homotopy equivalences to genuine equivariant homeomorphisms via surgery-type arguments. This advances rigidity results beyond tori to the non-commutative $(S^3)^n$-actions and clarifies how orbit-space combinatorics governs global equivariant topology.
Abstract
Quoric manifolds are the quaternionic analogue of toric manifolds. They admit a locally nice action of $(S^3)^n$ and the quotient is a manifold with corners. We show that they satisfy equivariant rigidity. More precisely, any locally linear $(S^3)^n$-manifold that it is equivariantly homotopic equivalent to a quoric manifold is equivariantly homeomorphic to it. The proof is given by generalising the methods of used in Coxeter and toric manifolds.