  v1: In this paper, we will give an elementary proof by the Heegaard
splittings of the 3-dimentional Poincare conjecture in point of view of PL
topology. This paper is of the same theory in [4](1983) excluding the last
three lines of the proof of the main theorem. v2: This paper gives the basic
result of [1](1997), i.e., a handle sliding and a band move of Heegaard
diagrams correspond to a replacement and a substitution in relations of the
fundamental groups derived from Heegaard diagrams, respectively (Theorem 12).
Corollary 13 is a new addition for the homotopy 3-sphere.

General Mathematics