Doctoral Dissertations

Date of Award

5-2018

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

Jerzy Dydak

Committee Members

Michael W. Berry, Nikolay Brodskiy, Morwen B. Thistlethwaite

Abstract

Coarse geometry has its roots in an attempt to make progress on the Novikov conjecture. It proved to be useful and resulted in progress on the Coarse Baum-Connes conjecture. This progress in turn led to progress in the Novikov conjecture. This paper investigates various constructions in coarse geometry that make new coarse geometric spaces from old ones.Chapter one is devoted to introductory material and builds an appropriate framework that we will use throughout the rest of this paper. Chapter two is about the asymptotic filtered colimit, a coarse construction that is intuitively a "coarse version" of the pasting lemma from Topology. We investigate many coarse properties that are (and are not) preserved under this construction.Chapter three is concerned with asymptotic products, a coarse construction that is an analog of the product topology. We investigate this construction in non-metrizable and metrizable settings. This chapter culminates with a result that, for certain circumstances, coarse embeddings are preserved in the asymptotic product construction.Chapter four is about coarse quotient mappings similar to the quotient mappings of Topology; we also introduce the coarse category in this chapter. We then go on to talk about coarse quotients by group actions. This leads us to consider a construction called warped spaces. Using warped spaces, we obtain results regarding the preservation of Property A under the warped space construction and (with certain additional assumptions) a "coarse deck transformation theorem".

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