{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/105627"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/105627","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Orderability of homology spheres obtained by Dehn filling","abstract":"In my thesis, I study left-orderability of $\\mathbb{Q}$-homology spheres. I use $\\widetilde{PSL_2\\mathbb{R}}$ representations as a tool. First, I showed this tool has its limitations by constricting a series of $\\mathbb{Z}$-homology spheres with potentially left-orderable fundamental groups but no non trivial $\\widetilde{PSL_2\\mathbb{R}}$ representations. However, this tool is still useful in most cases. With $\\widetilde{PSL_2\\mathbb{R}}$ representations, I construct the holonomy extension locus of a $\\mathbb{Q}$-homology solid torus which is an analog of its translation extension locus. Using extension loci, I study $\\mathbb{Q}$-homology 3-spheres coming from Dehn fillings of $\\mathbb{Q}$-homology solid tori and construct intervals of orderable Dehn fillings.","abstract_html":"In my thesis, I study left-orderability of $\\mathbb{Q}$-homology spheres. I use <span class=\"etd-inline-math\">\\widetilde{PSL<sub>2</sub>\\mathbb{R}}</span> representations as a tool. First, I showed this tool has its limitations by constricting a series of $\\mathbb{Z}$-homology spheres with potentially left-orderable fundamental groups but no non trivial <span class=\"etd-inline-math\">\\widetilde{PSL<sub>2</sub>\\mathbb{R}}</span> representations. However, this tool is still useful in most cases. With <span class=\"etd-inline-math\">\\widetilde{PSL<sub>2</sub>\\mathbb{R}}</span> representations, I construct the holonomy extension locus of a $\\mathbb{Q}$-homology solid torus which is an analog of its translation extension locus. Using extension loci, I study $\\mathbb{Q}$-homology 3-spheres coming from Dehn fillings of $\\mathbb{Q}$-homology solid tori and construct intervals of orderable Dehn fillings.","abstract_has_math":true,"creators":["Gao, Xinghua"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Dunfield, Nathan M.","Leininger, Christopher J.","Allen, Patrick B.","Bradlow, Steven B."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2019,"date_issued":"2019-11-26T20:33:46Z","date_published":"2019-11-26T20:33:46Z","updated_at":"2026-07-22T22:24:44Z","subjects":["left-orderable group","$\\widetilde{PSL_2\\mathbb{R}}$ representation"],"languages":["en"],"rights":["Copyright 2019 Xinghua Gao"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/105627","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Dunfield, Nathan M.","Leininger, Christopher J.","Allen, Patrick B.","Bradlow, Steven B."]},{"key":"dc:creator","label":"Author","values":["Gao, Xinghua"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2019-11-26T20:33:46Z","2019-07-03","2019-08"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["left-orderable group","$\\widetilde{PSL_2\\mathbb{R}}$ representation"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2019 Xinghua Gao"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/105627"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In my thesis, I study left-orderability of $\\mathbb{Q}$-homology spheres. I use $\\widetilde{PSL_2\\mathbb{R}}$ representations as a tool. First, I showed this tool has its limitations by constricting a series of $\\mathbb{Z}$-homology spheres with potentially left-orderable fundamental groups but no non trivial $\\widetilde{PSL_2\\mathbb{R}}$ representations. However, this tool is still useful in most cases. With $\\widetilde{PSL_2\\mathbb{R}}$ representations, I construct the holonomy extension locus of a $\\mathbb{Q}$-homology solid torus which is an analog of its translation extension locus. Using extension loci, I study $\\mathbb{Q}$-homology 3-spheres coming from Dehn fillings of $\\mathbb{Q}$-homology solid tori and construct intervals of orderable Dehn fillings.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2019-11-26 without embargo terms","The student, Xinghua Gao, accepted the attached license on 2019-06-30 at 00:17.","The student, Xinghua Gao, submitted this Dissertation for approval on 2019-06-30 at 00:54.","This Dissertation was approved for publication on 2019-07-03 at 10:46.","DSpace SAF Submission Ingestion Package generated from Vireo submission #14111 on 2019-11-26 at 12:50:18","Made available in DSpace on 2019-11-26T20:33:46Z (GMT). No. of bitstreams: 2 GAO-DISSERTATION-2019.pdf: 1264775 bytes, checksum: 83c0ef9199136d2a4c99ecfed00cdafe (MD5) LICENSE.txt: 4208 bytes, checksum: 0dba3989f7d21b66e16c3dcb1988bf38 (MD5) Previous issue date: 2019-07-03"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Orderability of homology spheres obtained by Dehn filling"]}]}],"canonical_facts":{"dc:contributor":["Dunfield, Nathan M.","Leininger, Christopher J.","Allen, Patrick B.","Bradlow, Steven B."],"dc:creator":["Gao, Xinghua"],"dc:date":["2019-11-26T20:33:46Z","2019-07-03","2019-08"],"dc:description":["In my thesis, I study left-orderability of $\\mathbb{Q}$-homology spheres. I use $\\widetilde{PSL_2\\mathbb{R}}$ representations as a tool. First, I showed this tool has its limitations by constricting a series of $\\mathbb{Z}$-homology spheres with potentially left-orderable fundamental groups but no non trivial $\\widetilde{PSL_2\\mathbb{R}}$ representations. However, this tool is still useful in most cases. With $\\widetilde{PSL_2\\mathbb{R}}$ representations, I construct the holonomy extension locus of a $\\mathbb{Q}$-homology solid torus which is an analog of its translation extension locus. Using extension loci, I study $\\mathbb{Q}$-homology 3-spheres coming from Dehn fillings of $\\mathbb{Q}$-homology solid tori and construct intervals of orderable Dehn fillings.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2019-11-26 without embargo terms","The student, Xinghua Gao, accepted the attached license on 2019-06-30 at 00:17.","The student, Xinghua Gao, submitted this Dissertation for approval on 2019-06-30 at 00:54.","This Dissertation was approved for publication on 2019-07-03 at 10:46.","DSpace SAF Submission Ingestion Package generated from Vireo submission #14111 on 2019-11-26 at 12:50:18","Made available in DSpace on 2019-11-26T20:33:46Z (GMT). No. of bitstreams: 2 GAO-DISSERTATION-2019.pdf: 1264775 bytes, checksum: 83c0ef9199136d2a4c99ecfed00cdafe (MD5) LICENSE.txt: 4208 bytes, checksum: 0dba3989f7d21b66e16c3dcb1988bf38 (MD5) Previous issue date: 2019-07-03"],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/105627"],"dc:language":["en"],"dc:rights":["Copyright 2019 Xinghua Gao"],"dc:subject":["left-orderable group","$\\widetilde{PSL_2\\mathbb{R}}$ representation"],"dc:title":["Orderability of homology spheres obtained by Dehn filling"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:44Z"}