Mark Walsh, National University of Ireland, Maynooth

Title: H-Spaces, Loop Spaces and Intermediate Ricci Curvatures.

Abstract: This is joint work with David Wraith. Over recent
years, a great deal has been discovered about the topology of the
space of Riemannian metrics of positive scalar curvature on various
smooth\ manifolds. In particular, when the underlying manifold in
question is a sphere, this space admits a natural H-space
multiplication based on the geometric connected sum technique of
Gromov and Lawson. Making use of the theory or operads and results of
Stasheff, Boardman, Vogt and May, it is possible to exhibit further
loop space structure. This was done in a paper of mine in 2014 and
there are various extensions and analogous results in more recent
papers by B. Botvinnik, J. Ebert, O. Randal Williams and G. Frenck.

In this talk, I will discuss a strengthening of these results, due to
David Wraith and myself for certain positive k-Ricci curvatures. These
curvatures, which were defined by Jon Wolfson for an n-dimensional
Riemannian manifold, are roughly a collection of partial ``averages"
of the eigenvalues of the Ricci tensor which interpolate between the
Ricci curvature (k=1) and the scalar curvature (k=n). Interestingly,
under reasonable conditions the above mentioned positive scalar
curvature results extend (via appropriate and delicate geometric
constructions) to hold for all positive k-Ricci curvatures between k=n
and k=2, only breaking down in the case of positive Ricci curvature (k=1).