Lower bounds for regular genus and gem-complexity of PL 4-manifolds

Within crystallization theory, two interesting PL invariants for $d$-manifolds have been introduced and studied, namely {\it gem-complexity} and {\it regular genus}. In the present paper we prove that, for any closed connected PL $4$-manifold $M$, its gem-complexity $\mathit{k}(M)$ and its regular genus $\mathcal G(M)$ satisfy: $$\mathit{k}(M) \ \geq \ 3 \chi (M) + 10m -6 \ \ \ \text{and} \ \ \ \mathcal G(M) \ \geq \ 2 \chi (M) + 5m -4,$$ where $rk(\pi_1(M))=m.$ These lower bounds enable to strictly improve previously known estimations for regular genus and gem-complexity of product 4-manifolds. Moreover, the class of {\it 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.Comment: 17 pages, 3 figures. To appear in Forum Mathematicu

A catalogue of orientable 3-manifolds triangulated by 30 coloured tetrahedra

The present paper follows the computational approach to 3-manifold classification via edge-coloured graphs, already performed in [1] (with respect to orientable 3-manifolds up to 28 coloured tetrahedra), in [2] (with respect to non-orientable3-manifolds up to 26 coloured tetrahedra), in [3] and [4] (with respect to genus two 3-manifolds up to 34 coloured tetrahedra): in fact, by automatic generation and analysis of suitable edge-coloured graphs, called crystallizations, we obtain a catalogue of all orientable 3-manifolds admitting coloured 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

3-manifolds efficiently bound 4-manifolds

It is known since 1954 that every 3-manifold bounds a 4-manifold. Thus, for instance, every 3-manifold has a surgery diagram. There are several proofs of this fact, including constructive proofs, but there has been little attention to the complexity of the 4-manifold produced. Given a 3-manifold M of complexity n, we show how to construct a 4-manifold bounded by M of complexity O(n^2). Here we measure ``complexity'' of a piecewise-linear manifold by the minimum number of n-simplices in a triangulation. It is an open question whether this quadratic bound can be replaced by a linear bound. The proof goes through the notion of "shadow complexity" of a 3-manifold M. A shadow of M is a well-behaved 2-dimensional spine of a 4-manifold bounded by M. We prove that, for a manifold M satisfying the Geometrization Conjecture with Gromov norm G and shadow complexity S, c_1 G <= S <= c_2 G^2 for suitable constants c_1, c_2. In particular, the manifolds with shadow complexity 0 are the graph manifolds.Comment: 39 pages, 21 figures; added proof for spin case as wel

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

Compact 3-manifolds via 4-colored graphs

We introduce a representation of compact 3-manifolds without spherical boundary components via (regular) 4-colored graphs, which turns out to be very convenient for computer aided study and tabulation. Our construction is a direct generalization of the one given in the eighties by S. Lins for closed 3-manifolds, which is in turn dual to the earlier construction introduced by Pezzana's school in Modena. In this context we establish some results concerning fundamental groups, connected sums, moves between graphs representing the same manifold, Heegaard genus and complexity, as well as an enumeration and classification of compact 3-manifolds representable by graphs with few vertices ($\le 6$ in the non-orientable case and $\le 8$ in the orientable one).Comment: 25 pages, 11 figures; changes suggested by referee: references added, figure 2 modified, results about classification of the manifolds in Proposition 17 announced at the end of section 9. Accepted for publication in RACSAM. The final publication is available at Springer (see DOI

Asymptotic behavior of quantum invariants

In this thesis we address the problem of the rate of growth of quantum invariants, specifically the Turaev-Viro invariants of compact manifolds and the related Yokota invariants for embedded graphs. We prove the recent volume conjecture proposed by Chen and Yang in two interesting families of hyperbolic manifolds. Furthermore we propose a similar conjecture for the growth of a certain quantum invariant of planar graphs, and prove it in a large family of examples. This new conjecture naturally leads to the problem of finding the supremum of the volume function among all proper hyperbolic polyhedra with a fixed 1-skeleton; we prove that the supremum is always achieved at the rectification of the 1-skeleton