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

Kramer, Linus

J. Differ. Geom. 64, No. 1, 1-55 (2003).

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.

Reviewer: Jonathan Hodgson (Philadelphia)

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

Keywords:

projective plane; Thom space; spherical bundle; Kirby-Siebenmann invariant; Pontrjagin numbers

Citations:

Zbl 0109.15701