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Algorithmic classification of 3-manifolds: Problems and results. (English. Russian original) Zbl 0981.57010

Bukhshtaber, V. M. (ed.) et al., Solitons, geometry, and topology: on the crossroads. Collected papers dedicated to the 60th birthday of Academician Sergei Petrovich Novikov. Transl. from the Russian. Moscow: MAIK Nauka/Interperiodica, Proc. Steklov Inst. Math. 225, 250-260 (1999); translation from Tr. Mat. Inst. Steklova 225, 264-275 (1999).
From the introduction: Classification of 3-manifolds is a key problem of three-dimensional topology. The solution to the problem of algorithmic classification is usually understood as an algorithmically constructed infinite list \(M_1\), \(M_2,\dots\) of manifolds in class \({\mathcal C}\) with the following two properties: (1) this list contains no duplicates, i.e., homeomorphic manifolds; (2) every manifold of class \({\mathcal C}\) is homeomorphic to one of the manifolds \(M_i\).
It should be mentioned that the algorithmic classification is a classification in a very weak sense. By analogy with the classification of surfaces or, say, simple finite groups, it is desirable to have a classifying list in an explicit form, for example, in the form of one or several parametric series of manifolds. Certain classes (for example, the Seifert manifolds), do admit such a classification; however, so far, there is no reason to believe that an explicit classification in the general case can be obtained. This paper is a survey of classification problems that have been solved or in the solution of which a considerable progress has been made.
For the entire collection see [Zbl 0967.00102].

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
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