Classical Lie group actions on \(\pi\)-manifolds
The purpose of this paper is to prove some nonexistence theorems of large group actions on certain manifolds. Let G be the classical group \(SU(m+1)\) or \(Sp(M+1)\), and accordingly denote by d the integer 2 or 4. Let \(M^{2dm-1}\) be a \((2dm-1)-\)dimensional compact connected \(\pi\)- manifold, \(m\geq 8\). Suppose that the first Pontryagin class vanishes and its \((dm-1)-\)dimensional integral homology group has a nontrivial cyclic subgroup of even or infinite order. Then the author proves that the manifold \(M^{2dm-1}\) does not admit a nontrivial G-action. Next let \(M^{2n-1}\) be a compact simply connected \((2n-1)-\)dimensional \(\pi\)-manifold. Suppose that \(n\geq 10\) and its \((n-1)-\)dimensional homology group is non trivial. Then the author proves that the manifold \(M^{2n-1}\) does not admit a nontrivial \(SO(n+1)\)-action with exceptions of the real Stiefel manifold \(V_{n+1,2}\) and a product manifold \(S^ n\times X^{n-1}\), where \(X^{n-1}\) is an \((n-1)-\)dimensional simply connected \(\pi\)-manifold without boundary. Further the author applies these results to group actions on sphere bundles over spheres.
group actions on sphere bundles over spheres
scientific article; zbMATH DE number 3856031