The Reconstruction and Realization of Topological Groups

Chang, Xiao (2018) The Reconstruction and Realization of Topological Groups. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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Abstract

A topological group is a group equipped with a topology so that the group operations are continuous. The symmetries (or automorphisms) of any given geometric object have a natural group structure, and often a canonical topology making them into a topological group.
Specifically, the automorphism groups of a countable structure is a topological group with the pointwise convergence topology, and the autohomeomorphism group of a compact space is a topological group with the compact-open topology. Here we investigate when these canonical topologies are minimal or the minimum amongst all Hausdorff group topologies.
Further, given an abstract topological group, we aim to realize it as the symmetry group of some geometric object with its canonical topology. We examine two such classes of objects: the automorphism groups of graphs with the pointwise convergence topology and the autohomeomorphism groups of continua (compact and connected spaces) with the compact-open topology.

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Details

Item Type:	University of Pittsburgh ETD
Status:	Unpublished
Creators/Authors:	Creators Email Pitt Username ORCID Chang, Xiao xic58@pitt.edu xic58
ETD Committee:	Title Member Email Address Pitt Username ORCID Committee Chair Gartside, Paul M. gartside@math.pitt.edu gartside Committee Member DeBlois, Jason C. jdeblois@pitt.edu jdeblois Committee Member Lennard, Christopher J. lennard@pitt.edu lennard Committee Member Szeptycki, Paul szeptyck@yorku.ca
Date:	26 September 2018
Date Type:	Publication
Defense Date:	30 July 2018
Approval Date:	26 September 2018
Submission Date:	22 July 2018
Access Restriction:	No restriction; Release the ETD for access worldwide immediately.
Number of Pages:	91
Institution:	University of Pittsburgh
Schools and Programs:	Dietrich School of Arts and Sciences > Mathematics
Degree:	PhD - Doctor of Philosophy
Thesis Type:	Doctoral Dissertation
Refereed:	Yes
Uncontrolled Keywords:	topological group, minimum group topology, minimal group topology, automorphism group, Cayley graph, locally finite graph, regular graph, rigid graph, autohomeomorphism group, profinite group, continuum, Cook continuum, rigid continuum.
Date Deposited:	26 Sep 2018 21:57
Last Modified:	26 Sep 2018 21:57
URI:	http://d-scholarship.pitt.edu/id/eprint/34963

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