The Poincaré Conjecture and related statements

The main topics of this paper are mathematical statements, results or problems related with the Poincaré conjecture, a recipe to recognize the threedimensional 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.