Loop structures on homotopy fibers of self maps of spheres

Let S^2n-1{k} denote the fiber of the degree k map on the sphere S^2n-1. If k = p^r, where p is an odd prime and n divides p - 1, then S^2n-1{k} is known to be a loop space. It is also known that S³{2^r} is a loop space for r ≥ 3. In this paper we study the possible loop structures on this family of spaces for all primes p. In particular we show that S³{4} is not a loop space. Our main result is that whenever S^2n-1{p^r} is a loop space, the loop structure is unique up to homotopy.