Free Actions on Cohomology Bazaikin Spaces
Date of Award
Doctor of Philosophy (PhD)
Mathematics | Physical Sciences and Mathematics
Bazaikin spaces are among the few known examples of Riemannian manifolds with positive sectional curvature. The main goal of this thesis is to explore finite groups that can act freely and isometrically on positively curved Riemannian manifolds with the same cohomology as Bazaikin spaces.
This thesis has two parts. The first part discusses free, smooth actions without any curvature and symmetry assumptionson the manifold. This is the main focus of Chapter 4. The goal here is to use an argument due to Heller to get obstructions on groups that can act freely.
In the second part, we get stronger obstructions by adding the assumptions of positive sectional curvature and torus symmetry. More specifically, we assume that the manifold admits an isometric action by a two-dimensional torus.
Along the way, we get some results about groups acting freely on another important family of positively curved Riemannian manifolds called Eschenburg spaces.
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Khalili Samani, Elahe, "Free Actions on Cohomology Bazaikin Spaces" (2021). Dissertations - ALL. 1413.