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).