The Poincaré conjecture and related statements. (English) Zbl 1454.57001
Dani, S. G. (ed.) et al., Geometry in history. Cham: Springer. 623-685 (2019).
Summary: 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.
For the entire collection see [Zbl 1426.01005].
For the entire collection see [Zbl 1426.01005].
MSC:
| 57-03 | History of manifolds and cell complexes |
| 01A60 | History of mathematics in the 20th century |
| 57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |
| 57K30 | General topology of 3-manifolds |
| 57R60 | Homotopy spheres, Poincaré conjecture |
| 57Q99 | PL-topology |
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 conjectureReferences:
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