{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/120241"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/120241","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Combinatorial complexes associated to surfaces","abstract":"Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2023-09-01 without embargo terms","abstract_html":"Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2023-09-01 without embargo terms","abstract_has_math":false,"creators":["Shinkle, Emily Suzanne"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Leininger, Christopher J","Dunfield, Nathan M","Albin, Pierre","Sadanand, Chandrika"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2023,"date_issued":"2023-05","date_published":"2023-05","updated_at":"2026-07-22T22:24:57Z","subjects":["Mapping Class Group","Flip Graph","Arc Complex","Rigidity","Finite Rigid Sets","Finite Rigidity","Geometric Topology","Geometric Group Theory","Arcs On Surfaces","Triangulations Of Surfaces"],"languages":["en","eng"],"rights":["Copyright 2023 Emily Shinkle"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2142/120241","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Leininger, Christopher J","Dunfield, Nathan M","Albin, Pierre","Sadanand, Chandrika"]},{"key":"dc:creator","label":"Author","values":["Shinkle, Emily Suzanne"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2023-05","2023-04-07"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"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":["Mapping Class Group","Flip Graph","Arc Complex","Rigidity","Finite Rigid Sets","Finite Rigidity","Geometric Topology","Geometric Group Theory","Arcs On Surfaces","Triangulations Of Surfaces"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en","eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2023 Emily Shinkle"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/2142/120241"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2023-09-01 without embargo terms","The student, Emily Shinkle, accepted the attached license on 2023-04-07 at 14:29.","The student, Emily Shinkle, submitted this Dissertation for approval on 2023-04-07 at 14:38.","This Dissertation was approved for publication on 2023-04-07 at 15:26.","DSpace SAF Submission Ingestion Package generated from Vireo submission #18932 on 2023-09-01 at 17:08:05","We study compact, connected, orientable surfaces, sometimes with boundary, and with a positive number of marked points such that there is at least one marked point on any boundary component. For surfaces without boundary and not isomorphic to a sphere with three marked points, we construct a finite rigid set of its arc complex: a finite simplicial subcomplex of its arc complex such that any locally injective map of this set into the arc complex of another surface with arc complex of the same or lower dimension is induced by a homeomorphism of the surfaces, unique up to isotopy in most cases. It follows that if the arc complexes of two surfaces are isomorphic, the surfaces are homeomorphic. We also give an exhaustion of the arc complex by finite rigid sets. This extends the results of Irmak-McCarthy. Further, we show that for most pairs of surfaces, possibly with boundary, there exists a finite subgraph of the flip graph of the first surface so that any injective homomorphism of this finite subgraph into the flip graph of the second surface can be extended uniquely to an injective homomorphism between the two flip graphs. Combined with a result of Aramayona-Koberda-Parlier, this implies that any such injective homomorphism of this finite set is induced by an embedding of the surfaces."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Combinatorial complexes associated to surfaces"]}]}],"canonical_facts":{"dc:contributor":["Leininger, Christopher J","Dunfield, Nathan M","Albin, Pierre","Sadanand, Chandrika"],"dc:creator":["Shinkle, Emily Suzanne"],"dc:date":["2023-05","2023-04-07"],"dc:description":["Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2023-09-01 without embargo terms","The student, Emily Shinkle, accepted the attached license on 2023-04-07 at 14:29.","The student, Emily Shinkle, submitted this Dissertation for approval on 2023-04-07 at 14:38.","This Dissertation was approved for publication on 2023-04-07 at 15:26.","DSpace SAF Submission Ingestion Package generated from Vireo submission #18932 on 2023-09-01 at 17:08:05","We study compact, connected, orientable surfaces, sometimes with boundary, and with a positive number of marked points such that there is at least one marked point on any boundary component. For surfaces without boundary and not isomorphic to a sphere with three marked points, we construct a finite rigid set of its arc complex: a finite simplicial subcomplex of its arc complex such that any locally injective map of this set into the arc complex of another surface with arc complex of the same or lower dimension is induced by a homeomorphism of the surfaces, unique up to isotopy in most cases. It follows that if the arc complexes of two surfaces are isomorphic, the surfaces are homeomorphic. We also give an exhaustion of the arc complex by finite rigid sets. This extends the results of Irmak-McCarthy. Further, we show that for most pairs of surfaces, possibly with boundary, there exists a finite subgraph of the flip graph of the first surface so that any injective homomorphism of this finite subgraph into the flip graph of the second surface can be extended uniquely to an injective homomorphism between the two flip graphs. Combined with a result of Aramayona-Koberda-Parlier, this implies that any such injective homomorphism of this finite set is induced by an embedding of the surfaces."],"dc:format":["application/pdf"],"dc:identifier":["https://hdl.handle.net/2142/120241"],"dc:language":["en","eng"],"dc:rights":["Copyright 2023 Emily Shinkle"],"dc:subject":["Mapping Class Group","Flip Graph","Arc Complex","Rigidity","Finite Rigid Sets","Finite Rigidity","Geometric Topology","Geometric Group Theory","Arcs On Surfaces","Triangulations Of Surfaces"],"dc:title":["Combinatorial complexes associated to surfaces"],"dc:type":["text","Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:57Z"}