Sasakian Manifolds with Perfect Fundamental Groups

TL;DR

This paper constructs numerous examples of higher-dimensional Sasakian manifolds with perfect fundamental groups, exhibiting extremal Sasaki metrics, including Sasaki-Einstein and Sasaki-η-Einstein types, expanding the known landscape of such geometries.

Contribution

It introduces a method to generate infinitely many Sasakian manifolds with perfect fundamental groups in all odd dimensions greater than one, featuring extremal and Einstein metrics.

Findings

Countably infinite examples in all odd dimensions >1

Existence of extremal Sasaki metrics with constant scalar curvature

Presence of Sasaki-Einstein and Sasaki-η-Einstein metrics

Abstract

Using the Sasakian join construction with homology 3-spheres, we give a countably infinite number of examples of Sasakian manifolds with perfect fundamental group in all odd dimensions greater than 1. These have extremal Sasaki metrics with constant scalar curvature. Moreover, we present further examples of both Sasaki-Einstein and Sasaki-$\eta$-Einstein metrics.