On small geometric invariants of 3-manifolds
If the \(2\)-spine of a \(3\)-manifold \(M\) has \(k\) edges, the corresponding Heegaard decomposition has \(k\) handles. The corresponding presentation of the fundamental group has \(k\) generators. It was conjectured that if the minimal genus Heegard decomposition of \(M\) was \(k\), then any presentation of the fundamental group of \(M\) would require at least \(k\) generators. \textit{M. Boileau} and \textit{H. Zieschang} showed this was not true. In [Invent. Math. 76, 455--468 (1984; Zbl 0538.57004)], they described a class of counterexamples. The author focuses attention on one of these counterexamples: a \(3\)-manifold \(M_1\) with fundamental group having a finite presentation with \(2\) generators despite \(M_1\) having Heegaard genus \(3\). In the absence of a spine of \(M_1\) corresponding to that \(2\) generator presentation, following Craggs one may seek an appropriate \(2\)-complex in \((M_1-(\text{3-cell})) \times [-1,1] \) carrying a presentation with \(2\) generators for the fundamental group of \(M_1\). The last 24 pages of the paper contain over 400 (mostly simple) equations and 17 increasingly detailed and impressive (and not so simple) diagrams which together describe the construction of such a \(2\) generator complex in \((M_1-(\text{3-cell})) \times [-1,1] \) from a \(3\) generator spine of \(M_1\).
scientific article; zbMATH DE number 5935412