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 = pr, where p is an odd prime and n divides p - 1, then S2n-1{k} is known to be a loop space. It is also known that S3{2r} 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 2n-1{pr} is a loop space, the loop structure is unique up to homotopy.