Branched Coverings of Manifolds and nH-Spaces

We show that on the sphere S m , m ≠ 1, 3, 7, there exists an n m -valued multiplication with unit for some n m ∈ {2, 4, 8}. We also explicitly construct a 2 k −1 -fold branched covering of S m 1 × ⋯ × S m k the product S m1 × ··· × S mk of k spheres over the sphere S m , m = m 1 + ··· + m k .