Exotic spheres with positive Ricci curvature

Abstract

We show that a certain class of manifolds admit metrics of positive Ricci curvature. This class includes many exotic spheres, including all homotopy spheres which represent elements of bP2n- §o.In this paper we investigate the Ricci curvature of a certain class of manifolds which includes many exotic spheres. In particular we will be concerned with constructing metrics of positive Ricci curvature. Our main result is as follows:Theorem 2.1. Homotopy spheres which bound parallelisable manifolds admit metrics of positive Ricci curvature.

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

“…It was shown in [18] that † arises as the boundary of a plumbed manifold: in fact † arises as the boundary of a plumbed manifold in infinitely many different ways. For this plumbed manifold, the disc-bundles involved are all D 2k -bundles over S 2k .…”

“…For example, twice a generator (2 ) is obtained from the following picture. Note that for each † 2 bP 4k , [18] establishes the existence (as opposed to the form) of corresponding simply connected plumbing diagrams with vertices representing D 2k -bundles over S 2k . It is easily checked (for example following [6, V.2]) that the graphs shown above have the required properties.…”

“…The positive scalar curvature metric on W means that the term ind D C .W; x g/ vanishes (see [ Note that the homotopy sphere † 2 bP 4k is a spin manifold with vanishing real Pontrjagin classes, and that the bounding manifold X p is shown to be parallelisable in [18].…”

“…The set of all homotopy spheres has a particularly nice and very large subfamily: the homotopy spheres which bound parallelisable manifolds. The author showed in [18] that these manifolds all admit Ricci positive metrics. (See also Boyer, Galicki and Nakamaye [5].…”

“…The key point is that a given homotopy sphere of this type can be realised as the boundary of infinitely many different plumbed manifolds. (For the basics of plumbing and applications to homotopy sphere construction see Browder [6, V.2] or Wraith [18].) Carr proves that for any regular neighbourhood of the union of base spheres in the plumbing construction, there is a metric of positive scalar curvature which is a product near the boundary, and thus the boundary itself has positive scalar curvature.…”