Some problems on 3-dimensional manifolds
C. D. Papakyriakopoulos · Bulletin of the American Mathematical Society · 1958
I. GENERALITIES 1. Introduction.One of the well-known problems in Topology is the classification problem of closed ^-dimensional manifolds.An n-manifold (^-dimensional manifold) is a connected separable metric space each of whose points has a closed neighborhood homeomorphic to a closed fz-cell.So we consider both manifolds with boundary and manifolds without boundary.A closed w-manifold is a compact w-manifold without boundary.Classification means to define an infinite sequence of closed nmanifolds Mi, Mi, M%, • • • , x such that any two of these are not homeomorphic, but any closed w-manifold M is homeomorphic with one of them.We emphasize that, we do not ask to find a method to decide with which of the model manifolds is M homeomorphic.We only want to know whether M is included in this sequence.Of course we do not ask to find an effective procedure, because such may not exist.The classification problem was solved long ago for n = 2, i.e. for closed surfaces, 2 [22, § §37-39, pp.130-142].So, as usual in Mathematics, one tries to solve the problem for the next dimension n = 3, in the hope that he will find a general method working for any dimension.This is the reason we restrict ourselves from now on to the case n = 3.The classification problem has been solved not only for closed surfaces, but also for compact nonclosed ones [22, §40, pp.142-144; 10, pp.151-158].See also [10, p. 171, 11. 12-16].We concentrate our attention on the classification problem of closed 3-manifolds, and for the time being we do not consider the classification problem for nonclosed 3-manifolds, because this last problem seems to be much more complicated, see No. 21. Generalities.As is well known, the classification problem is solved for n = 2 by cutting the surface along simple 8 curves.So the question arises naturally: Can we solve the classification problem for An address delivered before the Annual Meeting of the Society in Cincinnati, Ohio, on January 30, 1958, by invitation of the Committee to Select Hour Speakers for Annual and Summer Meetings; received by the editors March 18, 1958.1 To define Mi means to give a model of Mi, i.e. a way of constructing Mi.2 Numbers in brackets refer to the bibliography at the end of the paper.8 I.e.without self-intersections.