The Poincaré Conjecture and Related Statements

Valerii N. Berestovskii ()
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Valerii N. Berestovskii: Sobolev Institute of Mathematics SB RAS

Chapter Chapter 17 in Geometry in History, 2019, pp 623-685 from Springer

Abstract: Abstract The main topics of this paper are mathematical statements, results or problems related with the Poincaré conjecture, a recipe to recognize the three-dimensional sphere. The statements, results and problems are equivalent forms, corollaries, strengthenings of this conjecture, or problems of a more general nature such as the homeomorphism problem, the manifold recognition problem and the existence problem of some polyhedral, smooth and geometric structures on topological manifolds. Examples of polyhedral structures are simplicial triangulations and combinatorial simplicial triangulations of topological manifolds; so appears the triangulation conjecture, more exactly, the triangulation problem. Examples of geometric structures are Riemannian metrics that are locally homogeneous or have constant zero, positive or negative sectional curvature; more general structures are intrinsic or geodesic metrics with curvature bounded above or/and below in the sense of A.D. Alexandrov or with nonpositive curvature in the sense of H. Busemann.

Keywords: Alexandrov space with curvature bounded above; Bing-Borsuk problem; Busemann G-space; Closed 3-manifold; Closed simply connected 4-manifold; Combinatorial triangulation; Cone; Connected sum; Fundamental group; Haken manifold; Heegaard splittings; Homeomorphism problem; Homotopy group; Intersection form; JSJ-splitting theorem; Manifold recognition problem; Mapping class group; (Non)PL-conjecture; (Non)smoothable manifold; (Non)triangulable manifold; Poincaré conjecture; Prime decomposition; Seifert-van Kampen theorem; Smooth manifold; Sphere; Splitting homomorphism; Suspension; Thurston geometrization conjecture; Topological manifold; Triangulation; Triangulation conjecture; 57M40; 57R60; 57M35; 57R65; 57R05; 57R10; 52B70; 53B20; 51Fxx (search for similar items in EconPapers)
Date: 2019
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-030-13609-3_17

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DOI: 10.1007/978-3-030-13609-3_17

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