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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.

MSC:

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

Citations:

Zbl 0109.15701
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