On fibering spheres by toruses

Abstract

In this note we answer a question which was recently asked by H. Hopf, namely whether there exist fiber decompositions of the w-dimensional sphere S n , or of the ^-dimensional Euclidean space E n , with an ^-dimensional torus T 8 (the product of s factors S 1 ) as fiber. The answers are given in Theorems 1 and 2 below. The fiber decompositions are understood in the sense of fiber bundle [l], 1 with the group of all homeomorphisms as structure group, that is, we are given three spaces X, F and M (the bundle, the fiber and the base space), and a map $:X->itf, the projection; and for every point p of M there is given a neighborhood V p of p and a homeomorphism of <t>~" 1 (V p ) with the product V P XF, such that 4>~l(q) is mapped onto qXF for every'point qzV p . The sets <t>~l(p) are called the fibers.