Rank and Genus of 3-manifolds

A Heegaard splitting of a closed orientable 3-manifold M is a decomposition of M into two handlebodies along a closed surface S. The genus of S is the genus of the Heegaard splitting. The Heegaard genus of M , which we denote by g(M), is the minimal genus over all Heegaard splittings of M . A Heegaard splitting of genus g naturally gives a balanced presentation of the fundamental group π1(M): the core of one handlebody gives g generators, and the compressing disks of the other handlebody give a set of g relators. The rank of M , which we denote by r(M), is the rank of the fundamental group π1(M), that is, the minimal number of elements needed to generate π1(M). By the relation between a Heegaard splitting and π1(M) above, it is clear that r(M) ≤ g(M). In the 1960s, Waldhausen asked whether r(M) = g(M) for all M ; see [7, 40]. This was called the generalized Poincaré Conjecture in [7], as the case r(M) = 0 is the Poincaré conjecture. In [3], Boileau and Zieschang found a Seifert fibered space with r(M) = 2 and g(M) = 3. Later, Schultens and Weidmann [35] showed that there are graph manifolds M with discrepancy g(M)− r(M) arbitrarily large. A crucial ingredient in all these examples is that the fundamental group of a Seifert fibered space has an element commuting with other elements and, for a certain class of Seifert fibered spaces, one can use this property to find a smaller generating set of π1(M) than