An Elementary Proof by the Heegaard Splittings of the 3-Dimentional Poincare Conjecture

TL;DR

This paper presents an elementary proof of the 3-dimensional Poincaré conjecture using Heegaard splittings within PL topology, building on previous work and clarifying the relationship between diagram moves and fundamental group relations.

Contribution

It offers a simplified proof approach for the Poincaré conjecture via Heegaard splittings and establishes new connections between diagram moves and fundamental group relations.

Findings

Elementary proof of the Poincaré conjecture using Heegaard splittings

Handle sliding and band move correspond to group relation modifications

New corollary for homotopy 3-spheres

Abstract

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.