Classical Lie Group Actions on π-Manifolds

Abstract

The purpose of this paper is to prove some non existence theorems of large group actions on certain 7r-manifolds. We write G for the classical group SU(m + l) or Sp(m + l)5 and accordingly denote by d the integer 2 or 4. Let M~ be a (2dm — 1) -dimensional compact connected 7r-manifold5 m^8. Suppose that the first Pontrjagin class vanishes and its (dim — 1) -dimensional integral homology group has a nontrivial cyclic subgroup of even or infinite order. Then we shall prove that the manifold M~ can not admit a nontrivial G-action (Theorem 3 in Section 3). Next, let M be a compact simply connected (2n — -dimensional 7r-manifold. Suppose that n^ lO and its (n — 1) -dimensional homology group is nonzero. Then we shall prove that the manifold M~ can not admit a nontrivial 50(72 + 1) -action with exceptions of the real Stiefel manifold Vn+li2 and a product manifold S xX~, where X~ is an O — 1) -dimensional simply connected 7r-manifold without boundary (Theorem 4 in Section 3). Further we shall apply these results to study group actions on sphere bundles over spheres (Corollaries to Theorems).