REVIEW OF: "Li Tao, Rank and genus of 3-manifolds, J. Am. Math. Soc. 26, No. 3, 777-829 (2013)". [DE061686069]

In the 1960’s,Waldhausen conjectured the equality, for each closed orientable 3-manifoldM, between the rank r(M) of the fundamental group 1(M) and the Heegaard genus g(M) of M ([Algebr. Geom. Topol. 32(2), 313-322 (1978; Zbl 0397.57007)]). Various counterexamples may be found in the literature: in particular, Boileau and Zieschang obtained a Seifert fibered space with r(M) = 2 and g(M) = 3 ([Invent. Math. 76, 455-468 (1984; Zbl 0538.57004)]), while Schultens and Weidman proved the existence of graph manifolds with discrepancy g(M) − r(M) arbitrarily large ([Pac. J. Math. 231 (2), 481-510 (2007; Zbl 1171.57020)]). In 2007, the above Conjecture has been re-formulated, by restricting the attention to hyperbolic 3-manifolds: see [Geometry and Topology Monographs 12, 335-349 (2007; Zbl 1140.57009)]. The present paper gives a negative answer to this “modern version” of Waldhausen Conjecture. In fact, a closed orientable hyperbolic 3-manifold M is proved to exist, so that g(M) − r(M) is arbitrarily large. Actually, the author produces an (atoroidal) 3-manifold with boundary ¯M with r( ¯M ) < g( ¯M), and the closed counterexample is constructed starting from ¯M , via the so called annulus sum (see [Geom. Topol. 14 (4), 1871-1919 (2010; Zbl 1207.57031)]). Moreover, every 2-bridge knot exterior is proved to be a JSJ piece of a closed 3-manifold M with r(M) < g(M).