Betti Numbers of 3-sasakian Quotients of Spheres by Tori

We give a formula for the Betti numbers of 3-Sasakian manifolds or orbifolds which can be obtained as 3-Sasakian quotients of a sphere by a torus. This answers a question of Galicki and Salamon about the topology of 3-Sasakian manifolds. A (4m+3)-dimensional manifold is 3-Sasakian if it possesses a Riemannian metric with three orthonormal Killing elds deening a local SU(2)-action and satisfying a curvature condition. A complete 3-Sasakian manifold S is compact and its metric is Einstein with scalar curvature 2(2m + 1)(2m + 3). Moreover the local action extends to a global action of SO(3) or Sp(1) and the quotient of S is a quaternionic KK ahler orbifold. A large family of compact non-homogeneous 3-Sasakian manifolds was found by Boyer, Galicki and Mann in BGM2]. They are obtained by the 3-Sasakian reduction procedure, analogous to the symplectic or hyperkk ahler quotient construction, from the standard (4m + 3)-sphere. Recently, in BGMR], Boyer, Galicki, Mann and Rees have calculated the second Betti number of a 7-dimensional 3-Sasakian quotient of the (4q + 7)-sphere by a torus, as being equal to q. Using the ideas from BD], we shall give a formula for the Betti numbers of 3-Sasakian quotients of spheres by tori, valid in arbitrary dimension.