Topology of quaternionic manifolds

Manifolds with holonomy groups in SO(n) are the oriented Riemannian manifolds. Only general results may be obtained about the topology of this large class. The cohomology of Riemannian manifolds with holonomy groups in U(n) (Kahler manifolds), has been extensively studied (see [3], [6], [13]). The existence of compact Riemannian manifolds with holonomy groups in SU(n) or Sp(n) is not known for n =A 1. Hence, for the general groups, the most interesting cases left are those manifolds whose holonomy groups form subgroups of Sp(n) x Sp(l). These manifolds are called quaternionic manifolds. In the first part of this paper (??1-3), a decomposition analogous to the Hodge Decomposition for Kaihler manifolds is given for quaternionic manifolds (Theorem 3.5). Using a theorem of Chern, we get an increasing sequence of Betti numbers (Theorem 3.6). In the last part (??4 and 5), we define a quaternionic pinching. Using it, we give a quaternionic analogue (Theorem 5.5) to Klingenberg's Kahler pinching in [7] and [8].