Author(s): He Baihe
Pages: 241-244
Year: 1993 Issue:  2
Journal: Journal of Mathematical Research and Exposition
Keyword:  Poincar; 英文; word; 3维流形; have; only; 同胚; positively; Monte; 数学系;
Abstract: 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 0, 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.