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Fifty years ago: Topology of manifolds in the 50’s and 60’s

2009; Linguagem: Inglês

10.1090/pcms/015/02

2472-5064

John Milnor,

Homotopy and Cohomology in Algebraic Topology

The 1950’s and 1960’s were exciting times to study the topology of manifolds. This lecture will try to describe some of the more interesting developments. The flrst two sections describe work in dimension 3, and in dimensions n ‚ 5, while x3 discusses why it is often easier to work in higher dimensions. The last section is a response to questions from the audience. 1 3-Dimensional Manifolds. A number of mathematicians worked on 3-dimensional manifolds in the 50’s. (I was certainly one of them.) But I believe that the most important contribution was made by just one person. Christos Papakyriakopoulos had no regular academic position, and worked very much by himself, concentrating on old and di‐cult problems. We were both in Princeton during this period, and I saw him fairly often, but had no idea that he was doing such important work. (In fact, I don’t really remember talking to him|perhaps we were both too shy.) Let me try to explain what he accomplished. In 1910, Max Dehn had claimed a proof of the following lemma: If a piecewise linear map from a 2-simplex ¢ into a triangulated 3manifold is one-to-one near @¢, and if the image of the interior is disjoint from the image of the boundary, then there exists a piecewise linear embedding of ¢ which agrees with the original map near @¢: