The Fundamental Group: A Geometric Perspective

<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>

Name thesis:degree_name

Master of Science

Level thesis:degree_level

Thesis

Discipline thesis:degree_discipline

Math and Computer Science

Year dc:date.available

2026

Author dc:creator

• Bittenbinder, Kayla M

Contributors dc:contributor

• Joshua Thompson

• Fundamental group
• loop
• homotopy
• covering space
• universal cover
• holonomy
• parallel transport
• metric
• isometry
• Algebra
• Geometry and Topology
• Mathematics
• Physical Sciences and Mathematics

Repository record dc:identifier

https://commons.nmu.edu/theses/920

OAI identifier oai:identifier

oai:commons.nmu.edu:theses-1978