On the moduli space of positive Ricci curvature metrics on homotopy spheres

We show that the moduli space of Ricci positive metrics on certain homotopy spheres has infinitely many connected components.Comment: 28 pages, 11 figures. The text has been substantially re-written to improve the expositio

Positive Ricci curvature on highly connected manifolds

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.Comment: Corrected some minor typos and changed document class to amsart. The new document class added 10 pages, so the paper is now now 46 page

A generalization of the Perelman gluing theorem and applications

We prove a gluing result that allows to glue two Riemannian manifolds of positive intermediate Ricci curvature along their boundaries, provided the boundaries are isometric and the sum of second fundamental forms is positive semi-definite. This holds in particular for positive sectional curvature and generalizes a result of Perelman for positive Ricci curvature. As application we derive a sufficient condition for the existence of a metric with positive intermediate Ricci curvature and totally geodesic boundary, and obtain results on the observer moduli space of metrics of positive intermediate Ricci curvature on the sphere.Comment: 21 page

On Ptolemaic metric simplicial complexes

We show that under certain mild conditions, a metric simplicial complex which satisfies the Ptolemy inequality is a CAT(0) space. Ptolemy’s inequality is closely related to inversions of metric spaces. For a large class of metric simplicial complexes, we characterize those which are isometric to Euclidean space in terms of metric inversions

Protein kinase C theta is required for efficient induction of IL-10-secreting T cells

<div><p>Secretion of interleukin-10 (IL-10) by CD4⁺ T cells is an essential immunoregulatory mechanism. The work presented here assesses the role of the signaling molecule protein kinase C theta (PKCθ) in the induction of IL-10 expression in CD4⁺ T cells. Using wildtype and PKCθ-deficient Tg4 T cell receptor transgenic mice, we implemented a well-described protocol of repeated doses of myelin basic protein (MBP)Ac1-9[4Y] antigen to induce Tr1-like IL-10⁺ T cells. We find that PKCθ is required for the efficient induction of IL-10 following antigen administration. Both serum concentrations of IL-10 and the proportion of IL-10⁺ T cells were reduced in PKCθ-deficient mice relative to wildtype mice following [4Y] treatment. We further characterized the T cells of [4Y] treated PKCθ-deficient Tg4 mice and found reduced expression of the transcription factors cMaf, Nfil3 and FoxP3 and the surface receptors PD-1 and Tim3, all of which have been associated with the differentiation or function of IL-10⁺ T cells. Finally, we demonstrated that, unlike [4Y] treated wildtype Tg4 T cells, cells from PKCθ-deficient mice were unable to suppress the priming of naïve T cells <i>in vitro</i> and <i>in vivo</i>. In summary, we present data demonstrating a role for PKCθ in the induction of suppressive, IL-10-secreting T cells induced in TCR-transgenic mice following chronic antigen administration. This should be considered when contemplating PKCθ as a suitable drug target for inducing immune suppression and graft tolerance.</p></div