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Quoric Manifolds
Hopkinson, Jeremy Franklin Lawrence
[Thesis]. Manchester, UK: The University of Manchester; 2012.
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Abstract
Davis and Januszkiewicz introduced in 1981 a family of compact real manifolds, the Quasi-Toric Manifolds, with a group action by a torus, a direct product of circle (T) groups. Their manifolds have an orbit space which is a simple polytope with a distinct isotropy subgroup associated to each face of the polytope, subject to some consistency conditions. They defined a characteristic function which captured the properties of the isotropy subgroups, and showed that their manifolds can be classified by the polytope and characteristic function. They further showed that the cohomology ring of the manifold can be written down directly from properties derived from the polytope and the characteristic function. This work considers the question of how far the circle group T can be replaced by the group of unit quaternions Q in the construction and description of quasi-toric manifolds. Unlike T, the group Q is not commutative, so the actions of Q^n on the product H^n of the set of quaternions using quaternionic multiplication are studied in detail. Then, in direct analogy to the quasi-toric manifolds, a family of compact real manifolds, the Quoric Manifolds, is introduced which have an action by Q^n, and whose orbit space is a polytope. A characteristic functor is defined on the faces of the polytope which captures the properties of the isotropy classes of the orbits of the action. It is shown that quoric manifolds can be classified in a manner similar to the quasi-toric manifolds, by the polytope and characteristic functor. A restricted family, the global quoric manifolds, which satisfy an additional condition are defined. It is shown that an infinite number of polytopes exist in any dimension over which a global quoric manifold can be defined. It is shown that any global quoric manifold can be described as a quotient space of a moment angle complex over the polytope, and that its integral cohomology ring can be calculated, taking a form analagous to that in the quasi-toric case.