We give examples of knots in a genus 2 handlebody which have nontrivial Dehn surgeries yielding handlebodies and show that these knots are not 1--bridge. / text

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Given a symplectic manifold, it is of interest how Lie group actions, their orbit spaces look like and what are some topological requirements on the existence of such actions. In this thesis we present the work of Ono, giving some sufficient conditions for non-existence of circle actions on symplectic manifolds and work of Li, describing the fundamental groups of symplectic reductions of circle actions.

This thesis is an investigation of simple closed geodesics, or geodesic knots, in hyperbolic 3-manifolds. / Adams, Hass and Scott have shown that every orientable finite volume hyperbolic 3-manifold contains at least one geodesic knot. The first part of this thesis is devoted to extending this result. We show that all cusped and many closed orientable finite volume hyperbolic 3-manifolds contain infinitely many geodesic knots. This is achieved by studying infinite families of closed geodesics limiting to an infinite length geodesic in the manifold. In the cusped manifold case the limiting geodesic runs cusp-to-cusp, while in the closed manifold case its ends spiral around a short geodesic in the manifold. We show that in the above manifolds infinitely many of the closed geodesics in these families are embedded. / The second part of the thesis is an investigation into the topology of geodesic knots, and is motivated by Thurston’s Geometrization Conjecture relating the topology and geometry of 3-manifolds.We ask whether the isotopy class of a geodesic knot can be distinguished topologically within its homotopy class. We derive a purely topological description for infinite subfamilies of the closed geodesics studied previously in cusped manifolds, and draw explicit projection diagrams for these geodesics in the figure-eight knot complement. This leads to the result that the figure-eight knot complement contains geodesics of infinitely many different knot types in the3-sphere when the figure-eight cusp is filled trivially. / We conclude with a more direct investigation into geodesic knots in the figure-eight knot complement. We discuss methods of locating closed geodesics in this manifold including ways of identifying their isotopy class within a free homotopy class of closed curves. We also investigate a specially chosen class of knots in the figure-eight knot complement, namely those arising as closed orbits in its suspension flow. Interesting examples uncovered here indicate that geodesics of small tube radii may be difficult to distinguish topologically in their free homotopy class.

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.

This thesis is a study of fundamental properties of Hilbert space, properties of linear manifold, and realizations of Hilbert space.

Thesis advisor: Ian Biringer / We are interested in proving the following statement: Given a 3-manifold M with boundary and a homeomorphism of the boundary f : ∂M → ∂M such that there is some power that extends to M, there is some k depending only on the genus g(∂M) and some l < k such that ƒᶩ extends to M. We will prove that the power needed to extend is not uniformly bounded with some examples, we will prove the statement is true if M is boundary incompressible and we will show that the general statement reduces to effectivising some technical results about pure homeomorphisms extending to compression bodies. / Thesis (PhD) — Boston College, 2020. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

In order to discuss topological properties of quaternionic toric 8-manifolds, we introduce the notion of an algebraic morphism in the category of toric spaces. We show that the classification of quaternionic toric 8-manifolds with respect to an algebraic isomorphism is finer than the oriented topological classification. We construct infinite families of quaternionic toric 8-manifolds in the same oriented homeomorphism type but algebraically distinct. To prove that the elements within each family are of the same oriented homeomorphism type, and that we have representatives of all such types of a quaternionic toric 8-manifold, we present and use a method of evaluating the first Pontrjagin class for an arbitrary quaternionic toric 8-manifold.

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Manifold learning algorithms have been shown to be useful for many applications of numerical analysis. Unfortunately, existing algorithms often produce noisy results, do not scale well, and are unable to benefit from prior knowledge about the expected results. We propose a new algorithm that iteratively discovers manifolds by preserving the local structure among neighboring data points while scaling down the values in unwanted dimensions. This algorithm produces less noisy results than existing algorithms, and it scales better when the number of data points is much larger than the number of dimensions. Additionally, this algorithm is able to benefit from existing knowledge by operating in a semi-supervised manner.