Abstract. We show that the fundamental theorem of immersion theory admits a Poincaré duality space analogue. Along the way, we obtain new homotopy theoretic proofs of the existence and uniqueness of the Spivak normal fibration of a closed Poincaré space. 1.

In this paper we classify manifolds which look like projective planes. More precisely, we consider 1-connected closed topological manifolds M with integral homology H•(M) ∼ = Z 3. A straight-forward application of Poincaré duality shows that for such a manifold there exists a number m ≥ 2 such that Hk(M) = Z, for k = 0,m,2m; in particular, dim(M) = 2m is even. It follows from Adams ’ Theorem on the Hopf invariant that m divides 8. We construct a family of topological 2m-manifolds M(ξ) which are Thom spaces of certain topological R m-bundles (open disk bundles) ξ over the sphere S m, for m = 2,4,8, and which we call models. This idea seems to go back to Thom and was exploited further by Shimada [53] and Eells-Kuiper [14]. A particular case is worked out in some detail in Milnor-Stasheff [46] Ch. 20. However, these authors used vector bundles instead of Rm-bundles. We will see that the non-linearity of Rm-bundles yields many more manifolds than the construction by Eells-Kuiper. In [14] p. 182, the authors expressed the hope that “the given combinatorial examples form a complete set [...] for n ̸ = 4”. Our results show that in dimension n = 2m = 16, their construction missed 27/28 of the (infinitely many) combinatorial and topological solutions, while