Trisections of compact PL 4-manifolds via colored triangulations

The notion of trisection of a smooth, oriented, closed 4-manifold was introduced by Gay and Kirby in 2016, generalizing the idea of Heegaard splitting in dimension 3. More recently, Bell, Hass, Rubinstein and Tillmann started the study of unbalanced trisections via (singular) triangulations, by introducing a costruction that, under suitable conditions, gives rise to a trisection. This approach brings the advantages of a combinatorial description and allows to algorithmically construct trisections and estimate their complexity, starting from singular triangulations. A colored triangulation is a triangulation endowed with a vertex-labelling by 5 colors which is injective on every 4-simplex; it is called simple if each pair of vertices share exactly one edge. By applying the above construction to simple colored triangulations, Spreer and Tillmann succeeded in computing the trisection genus of all standard closed simply-connected 4-manifolds. In this thesis, after introducing trisections and describing their relationships to handle decompositions and Kirby diagrams, we present a generalization of the definition of trisection, via colored triangulations, to compact PL 4-manifolds with connected boundary: each compact PL 4- manifold with empty or connected boundary is proved to admit a decomposition satisfying a weaker notion of trisection (quasi-trisection) that is induced by a colored triangulation. A condition has been obtained for the collapsibility of a particular 2-dimensional subcomplex of the first barycentric subdivision of the triangulation to a graph, guaranteeing that the quasi-trisection is actually a trisection. The condition is related to a presentation of the fundamental group of the singular manifold associated to the given 4-manifold. We describe an algorithmic way to construct and simplify this presentation, starting from an edge-colored graph dual to a colored triangulation of the manifold. We also present some interesting computational results obtained by running the implementation of this algorithm over a catalogue of edge-colored graphs representing closed PL 4- manifolds, up to 20 vertices. The last part of the thesis is dedicated to another IT topic: optimization and parallelization of a classification algorithm for the previous catalogue of graphs. The aim is to classify all the 4- manifolds represented by crystallizations with a fixed number of vertices, manipulating them through a particular sequence of combinatorial moves.