REVIEW OF: "Jaco William - Rubinstein J.Hyam - Tillmann Stephan, Z2-Thurston norm and complexity of 3-manifolds, Math. Ann. 356, No. 1, 1-22 (2013)". [DE061684636]

Given a closed, irreducible 3-manifold M (different from S3, RP2 and L(3, 1)), the complexity of M is known to be the minimum number of tetrahedra in a (pseudo-simplicial) triangulation of M: see [Acta Appl. Math. 19, No.2, 101-130 (1990; Zbl 0724.57012)] and [Algorithms and Computation in Mathematics 9, Springer (2007; Zbl 1128.57001)]. In [Algebr. Geom. Topol. 11(3), 1257-1265 (2011; Zbl 1229.57010)] the authors found a lower bound for the complexity of M, in case M having a connected double cover (or, equivalently, a non-trivial Z2-cohomology class). The present paper makes use of the notion of Z2-Thurston norm (an analogue of Thurston’s norm, defined in [Mem. Am. Math. Soc. 339, 99-130 (1986; Zbl 0585.57006)]) in order to obtain a new lower bound on the complexity of M, if M admits multiple Z2-cohomology classes. Moreover, the minimal triangulations realizing this bound are characterized, in terms of normal surfaces consisting entirely of quadrilateral discs. It is worthwhile to note that the combinatorial structure of a minimal triangulation turns out to be governed by 0-efficiency ([J. Differ. Geom. 65(1), 61-168 (2003; Zbl 1068.57023)]) and low degree edges ([J. Topol. 2(1), 157-180 (2009; Zbl 1227.57026)]), and that the unique minimal triangulation of the generalized quaternionic space S3/Q8k (8k 2 Z+) - already obtained by the authors via the first lower bound - actually realizes this new bound, too.