Loop structures on homotopy fibers of self maps of spheres

Let S2n-1{k} denote the fiber of the degree k map on the sphere S2n-1. If k = p(r), where p is an odd prime and n divides p - 1, then S2n-1{k} is know n to be a loop space. It is also known that S-3{2(r)} is a loop space for r greater than or equal to 3. In this paper we study the possible loop struc tures on this family of spaces for all primes p. In particular we show that S-3{4} is not a loop space. Our main result is that whenever S2n-1{p(r)} i s a loop space, the loop structure is unique up to homotopy.