An algorithmic approach to construct crystallizations of 33-manifolds from presentations of fundamental groups

We have defined weight of the pair $(\langle S \mid R \rangle, R)$ for a given presentation $\langle S \mid R \rangle$ of a group, where the number of generators is equal to the number of relations. We present an algorithm to construct crystallizations of 3-manifolds whose fundamental group has a presentation with two generators and two relations. If the weight of $(\langle S \mid R \rangle, R)$ is $n$ then our algorithm constructs all the $n$-vertex crystallizations which yield $(\langle S \mid R \rangle, R)$. As an application, we have constructed some new crystallizations of 3-manifolds. We have generalized our algorithm for presentations with three generators and certain class of relations. For $m\geq 3$ and $m \geq n \geq k \geq 2$, our generalized algorithm gives a $2(2m+2n+2k-6+\delta_n^2 + \delta_k^2)$-vertex crystallization of the closed connected orientable $3$-manifold $M\langle m,n,k \rangle$ having fundamental group $\langle x_1,x_2,x_3 \mid x_1^m=x_2^n=x_3^k=x_1x_2x_3 \rangle$. These crystallizations are minimal and unique with respect to the given presentations. If `$n=2$' or `$k\geq 3$ and $m \geq 4$' then our crystallization of $M\langle m,n,k \rangle$ is vertex-minimal for all the known cases.Comment: 24 pages, 8 figure

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

Nonorientable 3-manifolds admitting coloured triangulations with at most 30 tetrahedra

We present the census of all non-orientable, closed, connected 3-manifolds admitting a rigid crystallization with at most 30 vertices. In order to obtain the above result, we generate, manipulate and compare, by suitable computer procedures, all rigid non-bipartite crystallizations up to 30 vertices.Comment: 18 pages, 3 figure

A note about complexity of lens spaces

Within crystallization theory, (Matveev's) complexity of a 3-manifold can be estimated by means of the combinatorial notion of GM-complexity. In this paper, we prove that the GM-complexity of any lens space L(p,q), with p greater than 2, is bounded by S(p,q)-3, where S(p,q) denotes the sum of all partial quotients in the expansion of q/p as a regular continued fraction. The above upper bound had been already established with regard to complexity; its sharpness was conjectured by Matveev himself and has been recently proved for some infinite families of lens spaces by Jaco, Rubinstein and Tillmann. As a consequence, infinite classes of 3-manifolds turn out to exist, where complexity and GM-complexity coincide. Moreover, we present and briefly analyze results arising from crystallization catalogues up to order 32, which prompt us to conjecture, for any lens space L(p,q) with p greater than 2, the following relation: k(L(p,q)) = 5 + 2 c(L(p,q)), where c(M) denotes the complexity of a 3-manifold M and k(M)+1 is half the minimum order of a crystallization of M.Comment: 14 pages, 2 figures; v2: we improved the paper (changes in Proposition 10; Corollary 9 and Proposition 11 added) taking into account Theorem 2.6 of arxiv:1310.1991v1 which makes use of our Prop. 6(b) (arxiv:1309.5728v1). Minor changes have been done, too, in particular to make references more essentia

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

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

Complexity computation for compact 3-manifolds via crystallizations and Heegaard diagrams

The idea of computing Matveev complexity by using Heegaard decompositions has been recently developed by two different approaches: the first one for closed 3-manifolds via crystallization theory, yielding the notion of Gem-Matveev complexity; the other one for compact orientable 3-manifolds via generalized Heegaard diagrams, yielding the notion of modified Heegaard complexity. In this paper we extend to the non-orientable case the definition of modified Heegaard complexity and prove that for closed 3-manifolds Gem-Matveev complexity and modified Heegaard complexity coincide. Hence, they turn out to be useful different tools to compute the same upper bound for Matveev complexity.Comment: 12 pages; accepted for publication in Topology and Its Applications, volume containing Proceedings of Prague Toposym 201

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