Loop structures on homotopy fibers of self maps of spheres

Abstract

Let S 2 n -1 { k } denote the fiber of the degree k map on the sphere S 2 n -1 . If k = p r , where p is an odd prime and n divides p - 1, then S 2 n -1 { k } is known to be a loop space. It is also known that S 3 {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 S 3 {4} is not a loop space. Our main result is that whenever S 2 n -1 { p r } is a loop space, the loop structure is unique up to homotopy.