# Obstructions to free actions on Bazaikin spaces

## TL;DR

This paper investigates the fundamental group structure of positively curved manifolds with torus symmetry, focusing on those with universal covers sharing cohomology with Bazaikin spaces, a rare class of positively curved manifolds.

## Contribution

It provides new structural results for the fundamental group of such manifolds, expanding understanding of their topology under symmetry conditions.

## Key findings

- Fundamental groups are constrained by the presence of torus symmetry.
- Universal covers share cohomology with Bazaikin spaces.
- Results narrow the possible topologies of positively curved manifolds with symmetry.

## Abstract

Apart from spheres and an infinite family of manifolds in dimension seven, Bazaikin spaces are the only known examples of simply connected Riemannian manifolds with positive sectional curvature in odd dimensions. We consider positively curved Riemannian manifolds whose universal covers have the same cohomology as Bazaikin spaces and prove structural results for the fundamental group in the presence of torus symmetry