REVIEW of: "Wong Helen, Quantum invariants can provide sharp Heegaard genus bounds, Osaka J. Math. 48, No. 3 (2011), 709-717". [DE059690467]

It is well-known that the Heegaard genus g(M) of a 3-manifold M (i.e. the smallest integer so that M has a Heegaard splitting of that genus) is generally very difficult to compute. The present paper investigates the effectiveness of a lower bound on g(M) deriving from the Reshetikhin-Turaev invariants (see [Commun. Math. Phys. 121, No.3, 351-399 (1989; Zbl 0667.57005)] for a basic reference about the invariants, and [Topology 37, No.1, 219-224 (1998; Zbl 0892.57005)] and [Quantum invariants of knots and 3-manifolds, de Gruyter Studies in Mathematics 18, Walter de Gruyter, Berlin (1994; Zbl 0812.57003)] for the quoted bound). Until advent of the quantum invariants, the best known lower bound on g(M) was the rank of the fundamental group of M, r( 1M). In [Invent. Math. 76, 455-468 (1984; Zbl 0538.57004)], Boileau and Zieschang presented a particular set of Seifert fibered spaces M, with the relatively rare property that g(M) = 3, while r( 1M) = 2. By studying the examples of Boileau and Zieschang, the author proves that quantum invariants may be used to provide a lower bound on g(M) which is both simpler to calculate and strictly larger than r( 1M).