LOOP STRUCTURES ON HOMOTOPY FIBRES OF SELF MAPS OF A SPHERE

By Carles Broto and Ran Levi


Let $S^{2n-1}\{k\}$ denote the fibre 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 $S^3\{2^r\}$ is a loop space for $r\geq 3$. 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 $S^{2n-1}\{p^r\}$ i
a loop space, the loop structure is unique up to homotopy.