Primitive Roots of Unity in H-Manifolds

Abstract

Let (M be a group with unit element e. An element g C (M is a k-th root of unity, for some integer ki ? 2, if gk e. If, in addition, gi j e for all j 1,2, 2 , Ic-i, then g is a primitive k-th root. The purpose of this paper is to show that, when (M is a compact connected Lie group, the structure of primitive roots of unity is, to a considerable extent, determined by the topology of (M and hence is independent of the algebraic structure on (M. Our approach will be, instead of considering only Lie groups, to study a larger class of objects. By an H-manifold we shall mean a triple (M. m, e) where M is a compact connected topological manifold without boundary, e C M, and m: M X M -> M is a map such that mr(x, e) rm(e,x) ==x for all xC M. Define in1: M --VI by m, (x) ==x and, for 7A>2, define Mk(X) m (X, mk-l (X) ). A primitive 1-th root of unity in the H-manifold (M, m, e) is a point x C M such that mk(x) = e but mj (x) 7& e for all j < k. We will studyprimitive roots of unity in this setting. We remark that the subject is of interest, independent of its application to Lie groups. Recent results by Hilton, Belfi [1], Morgan [6], and Zabrodsky produce many new classes of H-manifolds which are not homeomorphic to any Lie group or even, in some cases, of the same homotopy type as any Lie group. We shall show that the existence of primitive roots of unity in an Hmanifold (M, m, e) is independent of the choice of the map m in the sense that: