The present paper follows the computational approach to 3-manifold classification via edge-colored graphs, already performed in [1] (with respect to orientable 3-manifolds up to 28 colored tetrahedra), in [2] (with respect to non-orientable 3-manifolds up to 26 colored tetrahedra), in [3] and [4] (with respect to genus two 3-manifolds up to 34 colored tetrahedra): in fact, by automatic generation and analysis of suitable edge-colored graphs, called crystallizations, we obtain a catalogue of all orientable 3-manifolds admitting colored triangulations with 30 tetrahedra. These manifolds are unambiguously identified via JSJ decompositions and fibering structures. It is worth noting that, in the present work, a suitable use of elementary combinatorial moves yields an automatic partition of the elements of the generated crystallization catalogue into equivalence classes, which turn out to be in one-to-one correspondence with the homeomorphism classes of the represented manifolds.

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In the present paper, we show that the only closed orientable 5 5 -manifolds of regular genus less or equal than seven are the 5 5 -sphere S 5 {\mathbb {S}^5} and the connected sums of m m copies of S 1 × S 4 {\mathbb {S}^1} \times {\mathbb {S}^4} , with m ⩽ 7 m \leqslant 7 . As a consequence, the genus of S 3 × S 2 {\mathbb {S}^3} \times {\mathbb {S}^2} is proved to be eight. This suggests a possible approach to the ( 3 3 -dimensional) Poincaré Conjecture, via the well-known classification of simply connected 5 5 -manifolds, obtained by Smale and Barden.

Within crystallization theory, two interesting PL invariants for d-manifolds have been introduced and studied, namely gem-complexity and regular genus. In the present paper we prove that, for any closed connected PL 4-manifold M , its gem-complexity k (M ) and its regular genus G(M ) satisfy:where rk(π 1 (M )) = m. These lower bounds enable to strictly improve previously known estimations for regular genus and gem-complexity of product 4-manifolds. Moreover, the class of semi-simple crystallizations is introduced, so that the represented PL 4-manifolds attain the above lower bounds. The additivity of both gem-complexity and regular genus with respect to connected sum is also proved for such a class of PL 4-manifolds, which comprehends all ones of "standard type", involved in existing crystallization catalogues, and their connected sums.MSC 2010 : Primary 57Q15. Secondary 57Q05, 57N13, 05C15.

The aim of this paper is twofold. On the one hand, it provides a review of the links between random tensor models, seen as quantum gravity theories, and the PL-manifolds representation by means of edge-colored graphs (crystallization theory). On the other hand, the core of the paper is to establish results about the topological and geometrical properties of the Gurau-degree (or G-degree) of the represented manifolds, in relation with the motivations coming from physics.\ud In fact, the G-degree appears naturally in higher dimensional tensor models as the quantity driving their 1/N expansion, exactly as it happens for the genus of surfaces in the two-dimensional matrix model setting.\ud In particular, the G-degree of PL-manifolds is proved to be finite-to-one in any dimension, while in dimension 3 and 4 a series of classification theorems are obtained for PL-manifolds represented by graphs with a fixed G-degree.\ud All these properties have specific relevance in the tensor models framework, showing a direct fruitful interaction between tensor models and discrete geometry, via crystallization theory