Positive Ricci curvature on connected sums of fibre bundles

TL;DR

This paper extends the construction of positive Ricci curvature metrics to connected sums of fibre bundles, demonstrating that core metrics can be lifted along fibre bundles and applying this to represent bordism classes.

Contribution

It introduces methods to lift core metrics along general fibre bundles, broadening the class of manifolds with positive Ricci curvature.

Findings

Core metrics can be lifted along fibre bundles.

Connected sums of fibre bundles admit positive Ricci curvature.

All torsion-free bordism classes can be represented by manifolds with positive Ricci curvature.

Abstract

We consider the problem of preserving positive Ricci curvature along connected sums. In this context, based on earlier work by Perelman, Burdick introduced the notion of core metrics and showed that the connected sum of manifolds with core metrics admits a Riemannian metric of positive Ricci curvature. He subsequently showed that core metrics exist on compact rank one symmetric spaces, certain sphere bundles and manifolds obtained as boundaries of certain plumbings. In this article, we show that core metrics can be lifted along general fibre bundles, including sphere bundles and projective bundles. Our techniques also apply to spaces that decompose as the union of two disc bundles such as the Wu manifold. As application we show that all classes in the torsion-free oriented bordism ring can be represented by connected manifolds of positive Ricci curvature.