Positive intermediate Ricci curvature on connected sums
Philipp Reiser, David J. Wraith
TL;DR
This work advances the study of positive intermediate Ricci curvature by extending the construction of Ric_k>0 metrics to connected sums and plumbings. It introduces k-core metrics as a robust generalization that guarantees the preservation of Ric_k>0 under connected sums, leveraging a Perelman-style neck construction and a gluing framework. A key contribution is a plumbing theorem that broadens the repertoire of manifolds (notably certain projective spaces) that admit Ric_k>0 on their boundaries, with explicit optimal k-values for HP^n and OP^2. The results imply that topological obstructions are sharp in this intermediate curvature setting and enable explicit constructions for sphere-bundle plumbings, providing new examples where Gromov-type Betti-number bounds fail for $Ric_k>0$. Overall, the paper enriches the toolkit for assembling manifolds with prescribed intermediate curvature through refined surgery, docking, and plumbing techniques.
Abstract
We consider the problem of performing connected sums in the context of positive $k^{th}$ intermediate Ricci curvature. We show that such connected sums are possible if the manifolds involved possess `$k$-core metrics' for some $k$. Here, a $k$-core metric is a generalization of the notion of core metric introduced by Burdick for positive Ricci curvature. Further, we show that connected sums of linear sphere bundles over bases admitting such metrics admit positive $k^{th}$ intermediate Ricci curvature for $k$ in a particular range. This follows from a plumbing result we establish, which generalizes other recent plumbing results in the literature and is possibly of independent interest. As an example of a manifold admitting a $k$-core metric, we prove that $\mathbb{H} P^n$ admits a $(4n-3)$-core metric and that $\mathbb{O}P^2$ admits a $9$-core metric, and we show that in both cases these are optimal.