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 π_{1}(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 π_{1}(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. |