Some Homotopy Properties of Topological Groups and Homogeneous Spaces

Because of the continuous group operations, a topological group has some peculiar homotopy properties. It is well-known that the fundamental group of a topological group is commutative, [4, p. 54].2 The higher homotopy groups of a topological group have been investigated by W. Ifurewicz [11], B. Eckmann [5], and others. According to J. H. C. Whitehead [13], an arcwise connected topological group is n-simple for each n > 1 in the sense of S. Eilenberg [6]. According to R. H. Fox [81], the torus homotopy groups of a topological group G are all commutative; then it follows that the Whitehead products, [14], between the elements of the homotopy groups of G are always zero. The object of the present paper is to generalize these results to the class of special homogeneous spaces, defined in ?3, including the spheres Sn as particular cases. With an aim to do this, we shall give in ?2 some additional homotopy properties of a topological group. Our main results3 are embodied in the theorems of ?3. Throughout this paper, let G be a locally connected topological group. Since the components of G are homeomorphic to each other, we assume G to be arcwise connected. Let II be an arewise connected closed subgroup of G. A mapping is a continuous transformation, and a homotopy is a continuous family of mappings ft, (0 < t ? 1).