PL 4-manifolds admitting simple crystallizations: framed links and regular genus

Simple crystallizations are edge-coloured graphs representing PL 4-manifolds with the property that the 1-skeleton of the associated triangulation equals the 1-skeleton of a 4-simplex. In the present paper, we prove that any (simply-connected) PL $4$-manifold $M$ admitting a simple crystallization admits a special handlebody decomposition, too; equivalently, $M$ may be represented by a framed link yielding $\mathbb S^3$, with exactly $\beta_2(M)$ components ($\beta_2(M)$ being the second Betti number of $M$). As a consequence, the regular genus of $M$ is proved to be the double of $\beta_2(M)$. Moreover, the characterization of any such PL $4$-manifold by $k(M)= 3 \beta_2(M)$, where $k(M)$ is the gem-complexity of $M$ (i.e. the non-negative number $p-1$, $2p$ being the minimum order of a crystallization of $M$) implies that both PL invariants gem-complexity and regular genus turn out to be additive within the class of all PL $4$-manifolds admitting simple crystallizations (in particular: within the class of all "standard" simply-connected PL 4-manifolds).Comment: 14 pages, no figures; this is a new version of the former paper "A characterization of PL 4-manifolds admitting simple crystallizations

Cataloguing PL 4-manifolds by gem-complexity

We describe an algorithm to subdivide automatically a given set of PL n-manifolds (via coloured triangulations or, equivalently, via crystallizations) into classes whose elements are PL-homeomorphic. The algorithm, implemented in the case n=4, succeeds to solve completely the PL-homeomorphism problem among the catalogue of all closed connected PL 4-manifolds up to gem-complexity 8 (i.e., which admit a coloured triangulation with at most 18 4-simplices). Possible interactions with the (not completely known) relationship among different classification in TOP and DIFF=PL categories are also investigated. As a first consequence of the above PL classification, the non-existence of exotic PL 4-manifolds up to gem-complexity 8 is proved. Further applications of the tool are described, related to possible PL-recognition of different triangulations of the K3-surface.Comment: 25 pages, 5 figures. Improvements suggested by the refere

Random Tensors and Quantum Gravity

We provide an informal introduction to tensor field theories and to their associated renormalization group. We focus more on the general motivations coming from quantum gravity than on the technical details. In particular we discuss how asymptotic freedom of such tensor field theories gives a concrete example of a natural "quantum relativity" postulate: physics in the deep ultraviolet regime becomes asymptotically more and more independent of any particular choice of Hilbert basis in the space of states of the universe.Comment: Section 6 is essentially reproduced from author's arXiv:1507.04190 for self-contained purpose of the revie

Simple crystallizations of 4-manifolds

Minimal crystallizations of simply connected PL 4-manifolds are very natural objects. Many of their topological features are reflected in their combinatorial structure which, in addition, is preserved under the connected sum operation. We present a minimal crystallization of the standard PL K3 surface. In combination with known results this yields minimal crystallizations of all simply connected PL 4-manifolds of "standard" type, that is, all connected sums of $\mathbb{CP}^2$, $S^2 \times S^2$, and the K3 surface. In particular, we obtain minimal crystallizations of a pair of homeomorphic but non-PL-homeomorphic 4-manifolds. In addition, we give an elementary proof that the minimal 8-vertex crystallization of $\mathbb{CP}^2$ is unique and its associated pseudotriangulation is related to the 9-vertex combinatorial triangulation of $\mathbb{CP}^2$ by the minimum of four edge contractions.Comment: 23 pages, 7 figures. Minor update, replacement of Figure 7. To appear in Advances in Geometr

Coloured graphs representing PL 4-manifolds

Crystallization theory is a representation method for compact PL manifolds by means of a particular class of edge-coloured graphs. The combinatorial nature of this representation allows to elaborate and implement algorithmic procedures for the generation and analysis of catalogues of closed PL n-manifolds. In this paper we discuss the concepts which are involved in these procedures for n = 4 and present classification results arising from the study of the initial segment of the catalogue