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M8130 Algebraic topology, tutorial 12, 2020 13. 1. 2021
Exercise 1. Show that (n — 1)-connected compact manifold of dim n is homotopy equivalent to Sn (n > 2).
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M8130 Algebraic topology, tutorial 12, 2020 \3. 1. 2021
Remark. We have got that a compact manifold 3-dim (compact) which is )t-connected is homotopy equivalent to S3. There is another result: Every 3-dim manifold simply connected compact manifold is homeomornhic to S3. This latter result (it might seem we are close to proving it) is actually famous Poincare conjecture, one of Millenium Prize Problems and it was already solved by Griqori Perelman in 2002. Interesting story and interesting mathematician for sure. Perelman declined Fields medal (among other prizes).
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