Riv. Mat. Univ. Parma, Vol. 2, No. 2, 2011

Sandro Buoncristiano and Francesco Mercuri

A brief history of the Poincaré conjecture

Pages: 209-254
Received: 2 March 2011   
Accepted: 24 May 2011
Mathematics Subject Classification (1991): 53C41, 53A10.

Keywords:Poincaré conjecture, h-cobordism Theorem, intersection form, Ricci flow.

Abstract: In 1904 H. Poincaré proposed the problem of deciding whether a closed simply connected 3-manifold exists which is not homeomorphic to the 3-sphere. A negative answer to the question was soon labeled as the Poincaré conjecture, although Poincaré himself made no attempt to answer it. The question puzzled mathematicians for over 100 years and finally a positive answer to the conjecture has recently been obtained by G. Perelman. As many of the problems remained open for a long time, Poincaré conjecture or, better, the attempt at a deeper understanding of the problem produced an incalculable amount of mathematics. The aim of this essay is to give a report, short and incomplete of necessity, of some of those developments as well as brief comments on still open problems.

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