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

Path-component invariants for spaces of positive scalar curvature metrics

The Kreck-Stolz s-invariant is a classic path-component invariant for the space and moduli space of positive scalar curvature metrics. It is an absolute (as opposed to relative) invariant, but this strength comes at the expense of being defined only under restrictive topological conditions. The aim of this paper is to construct an analogous invariant for certain product manifolds on which the s-invariant is not defined

Non-negative versus positive scalar curvature

We show that results about spaces or moduli spaces of positive scalar curvature metrics proved using index theory can typically be extended to non-negative scalar curvature metrics. We illustrate this by providing explicit generalizations of some classical results concerning moduli spaces of positive scalar curvature metrics. We also present the first examples of manifolds with infinitely many path-components of Ricci non-negative metrics in both the compact and non-compact cases

On G-manifolds with finitely many non-principal orbits

We consider a compact Lie group G acting smoothly on a compact manifold M. The cohomogeneity of such an action is the dimension of the space of orbits M/G

The fundamental group of non-negatively curved manifolds

The aim of this article is to offer a brief survey of an interesting, yet accessible line of research in Differential Geometry. A fundamental problem of mathematics is to understand the relationship between the geometry and topology of manifolds. The geometry of a manifold is determined by a Riemannian metric, that is, a smoothly varying inner product on the tangent bundle. Altering the Riemannian metric on a given manifold alters the way in which it curves. It is natural, therefore, to ask to what extent the possible curvatures of a manifold determine and are determined by its topology. Note that all manifolds are assumed to be Riemannian, smooth, complete and without boundary

Some new results in Ricci Curvature

The purpose of this note is to announce some new results (see [10]) concerning manifolds of positive Ricci curvature