何伯和.3维Poincaré猜想的一个证明(英文)[J].数学研究及应用,1993,13(2):241~244
3维Poincaré猜想的一个证明(英文)
A Proof of 3-dimensional Poincaré Conjecture
投稿时间:1991-08-05  
DOI:10.3770/j.issn:1000-341X.1993.02.015
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何伯和 吉林大学数学系 
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中文摘要:
      设M是一个连通闭3维流形而且π1(M)=1,…,xn;y1,…,yn>=1.在本文中,我们利用由π1(M)=1得出来的条件x1=y1a1y1-1…ykakyk-1(其中y
英文摘要:
      A Heegaard splitting of an orientable closed cnnected 3-manifold M is a closed connected surface F → M such that M is divided into two handlebodies. Let g(M) be the minimal genus of all such surfaces. Let r(M) be the rank of π1(M). Then r(M) ≤ g(M). Waldhausen ([3] p.320) asked whether r(M) = g(M) is true for all M. But Boileau and Zieschang gave a negative answer to the question by describing some Seifert manifold M with 2 = r(M) < g(M) = 3 ([4]). In this paper, however, we shall prove that if π1(M) is trivial, then g(M)=r(M), thus M has a Heegaard splitting with genus O,i.e. M is a 3-sphere. This is the assertion which Poincare' conjectured in 1904. There are to approaches to the Poincare' conjecture, but here we shall work on it through its Heegaard splitting.
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