Projective planes and their look-alikes. (English) Zbl 1068.57019

This paper classifies all closed topological manifolds with the integral homology of an orientable projective plane. In other words all manifolds for which \(H_{*}(M) \cong {\mathbb Z}^{3}\). Poincaré duality implies that these must be of the homotopy type of a projective plane. The classification finds more manifolds than were found by J. Eells and N. H. Kuiper [Publ. Math., Inst. Hautes Étud. Sci. 14, 181–222 (1962; Zbl 0109.15701)]. This is a result of working with spherical fibrations rather than vector bundles. A complete set of invariants for each class is given. The paper reviews much of the earlier work and so is largely self contained.


57N65 Algebraic topology of manifolds
57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
57Q25 Comparison of PL-structures: classification, Hauptvermutung
57R55 Differentiable structures in differential topology
55R25 Sphere bundles and vector bundles in algebraic topology


Zbl 0109.15701
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