A diffeomorphism classification of manifolds which are like projective planes

We give a complete diffeomorphism classification of 1-connected closed manifolds M with integral homology H∗(M) ∼= Z ⊕ Z ⊕ Z, provided that dim(M) 6= 4. The integral homology of an oriented closed manifold1 M contains at least two copies of Z (in degree 0 resp. dimM). IfM is simply connected and its homology has minimal size (i.e., H∗(M) ∼= Z⊕Z), then M is a homotopy sphere (i.e., M is homotopy equivalent to a sphere). It is well-known that any homotopy sphere of dimension n 6= 3 is homeomorphic to the standard sphere S of dimension n. By contrast, the cardinality of the set Θn of diffeomorphism classes of homotopy spheres of dimension n can be very large (but finite except possibly for n = 3, 4) [7]. In fact, the connected sum of homotopy spheres gives Θn the structure of an abelian group which is closely related to the stable homotopy group πn+k(S ), k ≫ n (currently known approximately in the range n < 100). Somewhat surprisingly, it is easier to obtain an explicit diffeomorphism classification of 1-connected closed manifolds whose integral homology consists of three copies of Z. Examples of such manifolds are the 1-connected projective planes (i.e., the projective planes over the complex numbers, the quaternions or the octonions). Eells and Kuiper pioneered the study of these ‘projective plane like’ manifolds [4] and obtained many important and fundamental results. For example, they show that the integral cohomology ring of such a manifold M is isomorphic to the cohomology ring of a projective plane, i.e., H∗(M) ∼= Z[x]/(x3). This in turn implies that the dimension of M must be 2m with m = 2, 4 or 8 (cf. [4, §5]). We remark that a 1-connected closed manifold M of dimension n ≥ 5 with H∗(M) ∼= Z⊕Z⊕Z admits a Morse function with three critical points, which is the assumption that Eells-Kuiper work with. Any 1-connected projective plane like manifold of dimension 4 is homeomorphic to the complex projective plane by Freedman’s homeomorphism classification of simply connected smooth 4-manifolds [5]. Eells and Kuiper prove that there are six (resp. sixty) homotopy types of projective plane like manifolds of dimension 2m for m = 4 (resp. m = 8) [4, §5]. They get close to obtaining a classification of these manifolds up to homeomorphism resp. diffeomorphism. One way to phrase their result is the following. If M is a smooth manifold of this type, This paper was supported by a research grant in the Schwerpunktprogramm Globale Differentialgeometrie by the Deutsche Forschungsgemeinschaft; the second author was partially supported by NSF grant DMS 0104077. All manifolds are assumed to be smooth.