We study an interplay between topology, geometry, and algebra. Topology is the study of properties unchanged by bending, stretching or twisting space. Geometry measures space through concepts such as length, area, and angles. In the study of two-dimensional surfaces one can go back and forth between picturing twists as either distortions of the geometric properties of the surface or as a wrinkling of the surface while leaving internal measures unchanged. The language of groups gives us a way to distinguish geometric structures. Understanding the mapping class group is an important and hard problem. This paper contributes to visualizing how the mapping class group acts on geometric structures. We explore the geometry of closed, compact, and orientable two-dimensional manifolds through direct visualization and computation. We prove that the mapping class group of a torus is isomorphic to SL2Z via direct matrix multiplication on the generating elements of the fundamental group. While the fundamental group of the torus has only one possible presentation, up to homeomorphism; the case for the genus 2 surface is more complicated. We prove that an octagon representing a genus 2 surface can have its edges identified in different combinations to produce exactly four different possible presentations of fundamental groups. We explore surgeries on one of those types and show that surgeries that preserve that type are equivalent to Dehn twists on the surface, which are generators of the mapping class group
","abstract_html":"<p>We study an interplay between topology, geometry, and algebra. Topology is the study of properties unchanged by bending, stretching or twisting space. Geometry measures space through concepts such as length, area, and angles. In the study of two-dimensional surfaces one can go back and forth between picturing twists as either distortions of the geometric properties of the surface or as a wrinkling of the surface while leaving internal measures unchanged. The language of groups gives us a way to distinguish geometric structures. Understanding the mapping class group is an important and hard problem. This paper contributes to visualizing how the mapping class group acts on geometric structures. We explore the geometry of closed, compact, and orientable two-dimensional manifolds through direct visualization and computation. We prove that the mapping class group of a torus is isomorphic to SL2<strong>Z</strong> via direct matrix multiplication on the generating elements of the fundamental group. While the fundamental group of the torus has only one possible presentation, up to homeomorphism; the case for the genus 2 surface is more complicated. We prove that an octagon representing a genus 2 surface can have its edges identified in different combinations to produce exactly four different possible presentations of fundamental groups. We explore surgeries on one of those types and show that surgeries that preserve that type are equivalent to Dehn twists on the surface, which are generators of the mapping class group</p>","abstract_has_math":false,"creators":["Clark, Andrea"],"institution":null,"degree_name":"Master of Science","degree_level":"Thesis","degree_discipline":"Math and Computer Science","degree_department":null,"school":null,"contributors":["Dr. Joshua Thompson"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2021,"date_issued":"2021-05-01T07:00:00Z","date_published":"2021-05-01T07:00:00Z","updated_at":"2026-07-24T03:24:17Z","subjects":["mapping class group","fundamental group","hyperbolic geometry","topology","torus","octagons","Geometry and Topology"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://commons.nmu.edu/theses/653","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Dr. Joshua Thompson"]},{"key":"dc:creator","label":"Author","values":["Clark, Andrea"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2021-04-02T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Math and Computer Science"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["mapping class group","fundamental group","hyperbolic geometry","topology","torus","octagons","Geometry and Topology"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://commons.nmu.edu/theses/653"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We study an interplay between topology, geometry, and algebra. Topology is the study of properties unchanged by bending, stretching or twisting space. Geometry measures space through concepts such as length, area, and angles. In the study of two-dimensional surfaces one can go back and forth between picturing twists as either distortions of the geometric properties of the surface or as a wrinkling of the surface while leaving internal measures unchanged. The language of groups gives us a way to distinguish geometric structures. Understanding the mapping class group is an important and hard problem. This paper contributes to visualizing how the mapping class group acts on geometric structures. We explore the geometry of closed, compact, and orientable two-dimensional manifolds through direct visualization and computation. We prove that the mapping class group of a torus is isomorphic to SL2Z via direct matrix multiplication on the generating elements of the fundamental group. While the fundamental group of the torus has only one possible presentation, up to homeomorphism; the case for the genus 2 surface is more complicated. We prove that an octagon representing a genus 2 surface can have its edges identified in different combinations to produce exactly four different possible presentations of fundamental groups. We explore surgeries on one of those types and show that surgeries that preserve that type are equivalent to Dehn twists on the surface, which are generators of the mapping class group
"]},{"key":"dc:title","label":"Title","values":["Visualizing Geometric Structures on Topological Surfaces"]}]}],"canonical_facts":{"dc:contributor":["Dr. Joshua Thompson"],"dc:creator":["Clark, Andrea"],"dc:date.available":["2021-04-02T07:00:00Z"],"dc:description.abstract":["We study an interplay between topology, geometry, and algebra. Topology is the study of properties unchanged by bending, stretching or twisting space. Geometry measures space through concepts such as length, area, and angles. In the study of two-dimensional surfaces one can go back and forth between picturing twists as either distortions of the geometric properties of the surface or as a wrinkling of the surface while leaving internal measures unchanged. The language of groups gives us a way to distinguish geometric structures. Understanding the mapping class group is an important and hard problem. This paper contributes to visualizing how the mapping class group acts on geometric structures. We explore the geometry of closed, compact, and orientable two-dimensional manifolds through direct visualization and computation. We prove that the mapping class group of a torus is isomorphic to SL2Z via direct matrix multiplication on the generating elements of the fundamental group. While the fundamental group of the torus has only one possible presentation, up to homeomorphism; the case for the genus 2 surface is more complicated. We prove that an octagon representing a genus 2 surface can have its edges identified in different combinations to produce exactly four different possible presentations of fundamental groups. We explore surgeries on one of those types and show that surgeries that preserve that type are equivalent to Dehn twists on the surface, which are generators of the mapping class group
"],"dc:identifier":["https://commons.nmu.edu/theses/653"],"dc:subject":["mapping class group","fundamental group","hyperbolic geometry","topology","torus","octagons","Geometry and Topology"],"dc:title":["Visualizing Geometric Structures on Topological Surfaces"],"thesis:degree_discipline":["Math and Computer Science"],"thesis:degree_level":["Thesis"],"thesis:degree_name":["Master of Science"]},"updated_at":"2026-07-24T03:24:17Z"}