Positive Ricci curvature on highly connected manifolds

TL;DR

The paper proves that certain highly connected manifolds, after connected sum with a homotopy sphere, admit metrics with positive Ricci curvature, using a new boundary description via explicit plumbings.

Contribution

It introduces a novel boundary description of these manifolds as explicit plumbings, enabling the construction of positive Ricci curvature metrics.

Findings

Existence of positive Ricci curvature metrics on these manifolds after connected sum.

New boundary description as explicit plumbings for highly connected manifolds.

Applicable to manifolds with specific connectivity and parallelizability conditions.

Abstract

For $k \ge 2,$ let $M^{4k-1}$ be a $(2k{-}2)$-connected closed manifold. If $k \equiv 1$ mod $4$ assume further that $M$ is $(2k{-}1)$-parallelisable. Then there is a homotopy sphere $\Sigma^{4k-1}$ such that $M \sharp \Sigma$ admits a Ricci positive metric. This follows from a new description of these manifolds as the boundaries of explicit plumbings.