This thesis explores the deep mathematical connection between the flexible, continuous world of topology and the rigid, distance-preserving world of metric geometry. We begin by constructing the fundamental group of a topological space, using the concept of loops and loop homotopy to identify a global topological invariant such as a hole or puncture. We then transition to the geometric realm of metric spaces and isometries, demonstrating how the isometry group of a space algebraically encodes its rigid symmetries. To bridge these two distinct mathematical frameworks, we utilize the construction of the universal covering space. We pull back the metric from a base space to its simply-connected universal cover to show that the covering projection acts as a local isometry. This allows us to analyze the symmetries of the cover through deck transformations. The paper culminates in a proof demonstrating that the fundamental group of a path-connected base space is naturally isomorphic to the group of deck transformations on its universal cover. Through concrete examples, we illustrate how this isomorphism successfully translates abstract topological features into concrete geometric symmetries. Finally, we extend our investigation from global symmetry to local curvature through the lens of parallel transport and the holonomy group. We establish a surjective homomorphism from the fundamental group to the quotient of the full holonomy group by the restricted holonomy group, demonstrating how the global topological features of a manifold strictly govern the geometric rotations induced by non-contractible loops.
","abstract_html":"<p>This thesis explores the deep mathematical connection between the flexible, continuous world of topology and the rigid, distance-preserving world of metric geometry. We begin by constructing the fundamental group of a topological space, using the concept of loops and loop homotopy to identify a global topological invariant such as a hole or puncture. We then transition to the geometric realm of metric spaces and isometries, demonstrating how the isometry group of a space algebraically encodes its rigid symmetries. To bridge these two distinct mathematical frameworks, we utilize the construction of the universal covering space. We pull back the metric from a base space to its simply-connected universal cover to show that the covering projection acts as a local isometry. This allows us to analyze the symmetries of the cover through deck transformations. The paper culminates in a proof demonstrating that the fundamental group of a path-connected base space is naturally isomorphic to the group of deck transformations on its universal cover. Through concrete examples, we illustrate how this isomorphism successfully translates abstract topological features into concrete geometric symmetries. Finally, we extend our investigation from global symmetry to local curvature through the lens of parallel transport and the holonomy group. We establish a surjective homomorphism from the fundamental group to the quotient of the full holonomy group by the restricted holonomy group, demonstrating how the global topological features of a manifold strictly govern the geometric rotations induced by non-contractible loops.</p>","abstract_has_math":false,"creators":["Bittenbinder, Kayla M"],"institution":null,"degree_name":"Master of Science","degree_level":"Thesis","degree_discipline":"Math and Computer Science","degree_department":null,"school":null,"contributors":["Joshua Thompson"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2026,"date_issued":"2026-05-01T07:00:00Z","date_published":"2026-05-01T07:00:00Z","updated_at":"2026-07-24T03:24:38Z","subjects":["Fundamental group","loop","homotopy","covering space","universal cover","holonomy","parallel transport","metric","isometry","Algebra","Geometry and Topology","Mathematics","Physical Sciences and Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://commons.nmu.edu/theses/920","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Joshua Thompson"]},{"key":"dc:creator","label":"Author","values":["Bittenbinder, Kayla M"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2026-04-03T07:00:00Z"]},{"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":["Fundamental group","loop","homotopy","covering space","universal cover","holonomy","parallel transport","metric","isometry","Algebra","Geometry and Topology","Mathematics","Physical Sciences and Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://commons.nmu.edu/theses/920"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["This thesis explores the deep mathematical connection between the flexible, continuous world of topology and the rigid, distance-preserving world of metric geometry. We begin by constructing the fundamental group of a topological space, using the concept of loops and loop homotopy to identify a global topological invariant such as a hole or puncture. We then transition to the geometric realm of metric spaces and isometries, demonstrating how the isometry group of a space algebraically encodes its rigid symmetries. To bridge these two distinct mathematical frameworks, we utilize the construction of the universal covering space. We pull back the metric from a base space to its simply-connected universal cover to show that the covering projection acts as a local isometry. This allows us to analyze the symmetries of the cover through deck transformations. The paper culminates in a proof demonstrating that the fundamental group of a path-connected base space is naturally isomorphic to the group of deck transformations on its universal cover. Through concrete examples, we illustrate how this isomorphism successfully translates abstract topological features into concrete geometric symmetries. Finally, we extend our investigation from global symmetry to local curvature through the lens of parallel transport and the holonomy group. We establish a surjective homomorphism from the fundamental group to the quotient of the full holonomy group by the restricted holonomy group, demonstrating how the global topological features of a manifold strictly govern the geometric rotations induced by non-contractible loops.
"]},{"key":"dc:title","label":"Title","values":["The Fundamental Group: A Geometric Perspective"]}]}],"canonical_facts":{"dc:contributor":["Joshua Thompson"],"dc:creator":["Bittenbinder, Kayla M"],"dc:date.available":["2026-04-03T07:00:00Z"],"dc:description.abstract":["This thesis explores the deep mathematical connection between the flexible, continuous world of topology and the rigid, distance-preserving world of metric geometry. We begin by constructing the fundamental group of a topological space, using the concept of loops and loop homotopy to identify a global topological invariant such as a hole or puncture. We then transition to the geometric realm of metric spaces and isometries, demonstrating how the isometry group of a space algebraically encodes its rigid symmetries. To bridge these two distinct mathematical frameworks, we utilize the construction of the universal covering space. We pull back the metric from a base space to its simply-connected universal cover to show that the covering projection acts as a local isometry. This allows us to analyze the symmetries of the cover through deck transformations. The paper culminates in a proof demonstrating that the fundamental group of a path-connected base space is naturally isomorphic to the group of deck transformations on its universal cover. Through concrete examples, we illustrate how this isomorphism successfully translates abstract topological features into concrete geometric symmetries. Finally, we extend our investigation from global symmetry to local curvature through the lens of parallel transport and the holonomy group. We establish a surjective homomorphism from the fundamental group to the quotient of the full holonomy group by the restricted holonomy group, demonstrating how the global topological features of a manifold strictly govern the geometric rotations induced by non-contractible loops.
"],"dc:identifier":["https://commons.nmu.edu/theses/920"],"dc:subject":["Fundamental group","loop","homotopy","covering space","universal cover","holonomy","parallel transport","metric","isometry","Algebra","Geometry and Topology","Mathematics","Physical Sciences and Mathematics"],"dc:title":["The Fundamental Group: A Geometric Perspective"],"thesis:degree_discipline":["Math and Computer Science"],"thesis:degree_level":["Thesis"],"thesis:degree_name":["Master of Science"]},"updated_at":"2026-07-24T03:24:38Z"}