Loop Structures on Homotopy Fibres of Self Maps of a Sphere Carlos Broto and Ran Levi

The problem of deciding whether or not a given topological space X is homotopy equivalent to a loop space is of classical interest in homotopy theory. Moreover, given that X is a loop space, one could ask whether the loop space structure is unique. A celebrated example of this is given in the classical work of Rector [23] where it is shown that the 3-sphere S admits infinitely many non-equivalent loop structures (see also [21]) and of Dwyer, Miller and Wilkerson [11], where the authors show that all such loop structures collapse to a single one if the sphere is localized at a prime p. The first problem is equivalent to the question whether given a space X, there exist a space BX and a homotopy equivalence of ΩBX to X. The second problem is equivalent to asking whether the homotopy type of the space BX is uniquely determined by the requirement that its loop space is homotopy equivalent to X. In this paper we study a family of spaces of classical interest in homotopy theory. Let S{d} denote the homotopy fibre of the degree d self map of S. Our aim is to study the possible loop structures supported by Sm{d} for various values of m and d. We restrict attention to the case where, d = 2. Our main theorem follows.