A counterexample to a conjecture of Whitehead and Volodin-Kuznetsov-Fomenko
Mitsuyuki Ochiai · Journal of the Mathematical Society of Japan · 1979
In the study of 3-manifolds, to contruct an algorithm of recognizing the standard 3-sphere $S^{3}$ among all 3-manifolds is a very important problem.The first basic work of this problem was done by Whitehead in 1936 [6], who discovered that certain (but not all) Heegaard diagrams for $S^{3}$ had a rather special geometric property (, see Conjecture A in the paper).Later Volodin- Kuznetsov-Fomenko conjectured that Heegaard diagrams for $S^{3}$ are reducible except for the canonical one.But Birman states in [2] that "nobody has succeeded in verifying such an assertion between 1935 and 1977, or producing a counterexample".Most recently Homma-Ochiai-Takahashi [3] proved that the conjecture is really true for the case of genus two.But in this paper we give a counterexample for the case of genus four.The Volodin-Kuznetsov- Fomenko-Whitehead algorithm is closely related with the algorithm to deter- mine whether a knot is trivial or not and so our counterexample is constructed as a branched covering space over a trivial 5-bridge knot.