Surgery on Ricci positive manifolds

Abstract

We consider performing surgery on Riemannian manifolds with positive Ricci curvature. We nd conditions under which the resulting manifold also admits a positive Ricci curvature metric. These conditions involve dimension and the form taken by the metric in a neighbourhood of the surgery.

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

“…Using this isometry as the trivialisation of the normal bundle, they show that the manifold resulting from the surgery has positive Ricci curvature provided the dimension of the surgery is at least one, the codimension at least three, and the ratio of the radius of the sphere to the radius of the disk is sufficiently small. This result was developed further in [19] where it was shown that under the same metric assumptions, Ricci positivity can be preserved if different trivialisations of the normal bundle are used. This generalisation comes at the expense of tighter dimensional requirements, but being able to work with different trivialisations of the normal bundle is vital for applying the results to the construction of homotopy spheres.…”

“…The statements we give below are slightly different to those presented in [19], so as to emphasize the metric features which we will need in the sequel. Theorem 3.2 Given E D D n S m , a constant 2 .0; 1/ and a choice of connection r , there is a constant 0 2 .0; 1/, 0 D 0 .n; m; ; r/, and a constant R 0 2 .0; 1=4/ independent of all parameters, such that for any choice of 0 Ä 0 there exists a number a D a.n; m; ; 0 / > 2 depending smoothly on 0 , with a !…”

“…In the remainder of this section, we summarise the Ricci positive surgery results of [19]. The statements we give below are slightly different to those presented in [19], so as to emphasize the metric features which we will need in the sequel.…”

“…For the reader wishing to compare the above Theorem with the paper [19], a few words of explanation are in order here. The smooth dependence of f , h and a on 0 might suggest that they are all uniquely determined by 0 (given n, m and r ).…”

“…However this is not the case, as certain choices have to be made during the construction of these objects. On the other hand, it is clear from the construction details in [19] that the relevant choices can be made so that the variation with respect to 0 is smooth.…”