Discrepancy between the rank and the Heegaard genus of a 3-manifold. (Spanish: La discrepancia entre el rango y el genero de Heegaard de una 3-variedad)

Abstract

Let M be a closed orientable 3-manifold. The rank of M, rk(M), is the minimum number of elements that can generate π1(M). Clearly rk(M)≤Hg(M), where Hg(M) is the Heegaard genus of M. F. Waldhausen conjectured that equality holds here, but then. Boileau and H. Zieschang [Invent. Math. 76 (1984), no. 3, 455–468;] found an infinite set of (Seifert) manifolds M with Hg(M)=3 and rk(M)=2. To prove the latter equality they started with the presentation of π1(M) resulting from a Heegaard splitting of M of genus 3; they then reduced this presentation to a presentation with only two generators. In the present paper the author shows that one can perform such a reduction by using only Nielsen moves (which are in general not sufficient for transforming an arbitrary finite presentation to any other finite presentation of the same group). Actually he proves a more general theorem about reducing, by Nielsen moves alone, the number of generators in certain presentations coming from Heegaard splittings, and because of this theorem he believes that the strict inequality rk(M)<Hg(M) is quite commonplace among closed orientable 3-manifolds.