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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.

Related project(s):
52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

We prove a canonical variation-type result for submersion metrics with positive intermediate Ricci curvatures. This can then be used in conjunction with surgery techniques to establish the existence of metrics with positive intermediate Ricci curvatures on a wide range of examples which had previously only been known to admit positive Ricci curvature, such as highly connected manifolds and exotic spheres. Further, we extend results of the second author on the moduli space of metrics with positive Ricci curvature to positive intermediate Ricci curvatures.

Related project(s):
52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

We express the total space of a principal circle bundle over a connected sum of two manifolds in terms of the total spaces of circle bundles over each summand, provided certain conditions hold. We then apply this result to provide sufficient conditions for the existence of free circle and torus actions on connected sums of products of spheres and obtain a topological classification of closed, simply connected manifolds with a free cohomogeneity-four torus action. As a corollary, we obtain infinitely many manifolds with Riemannian metrics of positive Ricci curvature and isometric torus actions.

Related project(s):
52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

The twisted suspension of a manifold is obtained by surgery along the fibre of a principal circle bundle over the manifold. It generalizes the spinning operation for knots and preserves various topological properties. In this article, we show that Riemannian metrics of positive Ricci curvature can be lifted along twisted suspensions. As an application we show that the maximal symmetry rank of a closed, simply connected Riemannian manifold of positive Ricci curvature is $(n-2)$ in all dimensions $n\geq 4$. Further applications include simply connected 6-manifolds whose homology has torsion, (rational) homology spheres in all dimensions at least 4, and manifolds with prescribed third homology.

Related project(s):
52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

We extend a positive Ricci curvature gluing theorem of Perelman to a range of positive intermediate curvature conditions, ranging from positive scalar curvature up to (and including) positive sectional curvature. As an application of this, we demonstrate that the observer moduli space of metrics with positive intermediate Ricci curvatures can have non-trivial higher homotopy groups. Further applications include deriving a sufficient condition for the existence of a metric with positive intermediate Ricci curvature and totally geodesic boundary.

Related project(s):
52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II