Note on a result of Boileau-Zieschang.

Abstract

The author starts with a discussion of various concepts of genus of a closed, orientable 3-manifold M including the well-known Heegaard genus, Hg(M), the rank, rk(M) (i.e. the minimum number of elements of π1(M) which suffice to generate π1(M), and the recently defined by Craggs extended Nielsen genus, EN(M). Next, he proposes still another invariant in the same vein. He calls the big genus of M, Bg(M), the smallest possible integer r, such that, if M∗ is M minus the interior of a 3-ball, M∗×I has a handle decomposition with just one 0-handle and the same number r of 1- and 2-handles (and no 3- and 4-handles). The result of Boileau-Zieschang referred to in the title asserts that Hg(M)=3 and rk(M)=2 for some Seifert manifolds M. Replacing the well-known conjecture that Hg(M)=rk(M) for all M, disproved by this result, the author suggests the following problem: Is rk(M)=Bg(M), for every M? Attempting to understand relations between all these genuses, the author studies a specific Seifert manifold M and proves that Hg(M)=3 and EN(M)=2 for this M. In the last section he tries, unsuccessfully, to compute Bg(M) for this M. The expected value of Bg(M) is 3. The main part of the paper is a sequence of twelve pictures.