Two New Jï-spaces by Morton Curtis and Guido Mislin

Two of these homotopy types are known 2ï-spaces; namely, SU(3)XS and 5 £7(4). The other two are the new ü-spaces of the title, and a word is in order as to in what sense they are new. If one seeks differentiable manifolds which are H-spaces not homeomorphic to known iT-spaces, then recent work of Belfi [ l ] and Morgan [4] furnish a big supply. For example, there are infinitely many nonhomeomorphic manifolds having the homotopy type of 5Z7(4) (and hence being ü-spaces). If one seeks new homotopy types (excluding, of course, cartesian products of known ones) the picture is quite different. Classically one knew only S and its projective space P 7 , except for Lie groups. In 1968 Hilton and Roitberg [2], [3] discovered a new JET-space, a principal S bundle over S. In 1969 Stasheff [5] found two more new ff-spaces among the seven homotopy types of principal S bundles over S. Our two new spaces brings the known total to seven in dimension :g 15. We have also shown that the three new homotopy types introduced by going from principal 5 bundles over S to 50(4) 3-sphere bundles over S are not if-spaces. The first part of our theorem is proved using the technique of mixing homotopy types (relative to a subdivision of the set of prime numbers) due to Zabrodsky [7], in much the same manner as Stasheff [5]. The second part uses the Adams operations in K-theory and a result of Suter [6] to distinguish the homotopy types.