Topological classification of Bazaikin spaces

TL;DR

This paper provides a topological classification of 13-dimensional manifolds modeled on Bazaikin spaces, extending the classification efforts from 7-dimensional Aloff-Wallach spaces to higher dimensions.

Contribution

It offers the first complete topological classification of Bazaikin spaces in dimension 13, building on previous classifications of similar positively curved manifolds.

Findings

Complete topological classification of 13-dimensional Bazaikin spaces

Extension of classification methods from 7D to 13D

Identification of topological invariants for Bazaikin spaces

Abstract

Geometry of manifolds with positive sectional curvature has been a central object dates back to the beginning of Riemannian geometry. Up to homeomorphism, there are only finitely many examples of simply connected positively curved manifolds in all dimensions except in dimension 7 and 13, namely, the Aloff-Wallach spaces and the Eschenburg spaces in dimension 7, and the Bazaikin spaces in dimension 13. The topological classification modelled on the 7-dimensional examples has been carried out by Kreck-Stolz which leads to a complete topological classification for the Aloff-Wallach spaces. The main goal of this paper is to provide the topological classification of 13-dimensional manifolds modelled on the Bazaikin spaces.