Automorphisms of 3-manifolds

University Of Oklahoma Norman Campus, Norman OK

Investigators

Abstract

Abstract Award: DMS-0102463 Principal Investigator: Darryl McCullough The proposed work advances a number of research projects in the area of low-dimensional topology. Their unifying theme is the automorphisms of 3-dimensional manifolds, including homotopy equivalences, diffeomorphisms, and isometries. Specific projects include: the Generalized Smale Conjecture for elliptic 3-manifolds, the isomorphism problem for diffeomorphism groups of elliptic 3-manifolds, generalization of the Abikoff-Maskit structure theory for Kleinian groups using topological methods, investigation of free actions of finite groups on orientable handlebodies using Nielsen equivalence classes of generating sets, and fibration theorems for spaces of fiber-preserving diffeomorphisms of manifolds having fiberings and singular fiberings. The primary mathematical constructs that will be investigated are 3-manifolds, which are geometric objects locally modeled on the 3-dimensional spatial structure of the physical universe, and groups, which are algebraic systems with an operation akin to the addition of ordinary numbers. Some of the ongoing work has already been applied in the theoretical physics of gravitation, but most of its applications are entirely within pure mathematics. The guiding philosophy of most of the research is to use topological and geometric structure of 3-manifolds to understand groups of symmetries and other kinds of automorphisms. Groups of automorphisms of a mathematical object often exhibit their own interesting structure. A classic example of this is the finite-dimensional vector spaces. They are rather simple objects, but their automorphism groups, the general linear groups, have a rich structure and find wide-ranging uses in mathematics and physics. Within the proposed work, an example is the 3-manifolds called handlebodies. These are among the simplest 3-manifolds to describe topologically, but their groups of symmetries are subtle and varied. In fact, any finite group can be a group of symmetries of some handlebody, and the number of distinct ways that a group can act as symmetries on a given handlebody can be quite large. A different use of the philosophy involves Kleinian groups, which are discrete groups of symmetries of 3-dimensional hyperbolic space. Each Kleinian group produces a quotient 3-manifold, and one of the projects uses the topological structure of these quotient 3-manifolds to give an algebraic classification of Kleinian groups.