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Combinatorics of triangulations of 3-manifolds. (English) Zbl 0785.57007

It is somewhat surprising to see that the main result of this paper is well known for more than 20 years. The list of references is rather poor; it mentions only one research paper. The subject is the following: For a given triangulation of a closed 3-manifold let \(f_ i\) denote the number of \(i\)-dimensional simplices. The Dehn-Sommerville equations say that \(f_ 0-f_ 1+f_ 2-f_ 3=0 =f_ 2-2f_ 3\). Therefore, only two of these four quantities are independent. The further investigation of inequalities between the \(f_ i\) is a classical subject in the theory of convex polytopes. In particular, for 3-manifolds one has \(f_ 1\geq 4f_ 0-10\) (or \(\mu_ 1\geq 9/2\) in the authors’ notations) with equality if and only if the manifold is a 3-sphere triangulated as the boundary of a stacked 4-polytope (= \(\#^ k\partial\Delta^ 4\) in the authors’ notations). This is known as the lower bound theorem. If the manifold is topologically distinct from the 3-sphere the inequality is sharper \(f_ 1\geq 4f_ 0\) (or \(\mu_ 0\geq 9/2\) in the authors’ notations) with equality only for 1-handles \(S^ 1\times S^ 2\) or \(S^ 1\underline{\times} S^ 2\) triangulated as stacked spheres after identification of two tetrahedra. The triangulations of this latter type are known in the literature as the class \({\mathcal H}^ 4(1)\).
All these results can be found in D. W. Walkup [Acta Math. 125, 75- 107 (1970; Zbl 0204.563)]. They constitute also the main theorem 3 of the paper under review. However, this paper is interesting because it adds a few ideas and aspects which seem to be neglected so far. As an example, the “average edge order” used by the authors is interesting with regard to positive or negative PL curvature. The “crowded triangulations” defined by the authors seem to be a new concept.
Unfortunately, there is not enough space in a review to cover the references missed in this paper. We just mention that the case of 4- manifolds is treated in the paper by D. Walkup (loc. cit.). Higher dimensional generalizations have been given by G.Kalai [Invent. Math. 88, 125-151 (1987; Zbl 0624.52004)]. Lower bounds for the number of vertices of triangulated manifolds have been obtained by U. Brehm and the reviewer [Topology 26, 465-473 (1987; Zbl 0681.57009)] for a survey see [the reviewer, “Advances in differential geometry and topology” (F. Tricerri, ed.). pp. 59-114, World Singapore 1990].

MSC:

57Q15 Triangulating manifolds
57N10 Topology of general \(3\)-manifolds (MSC2010)
52B70 Polyhedral manifolds
57M15 Relations of low-dimensional topology with graph theory
52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
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