Finding Nice Crystallizations for Handle Free $N$-Manifolds

Abstract

The paper deals with combinatorial structures (pseudo-complexes, crystallizations) giving a direct link between the topology of triangulated manifolds and the theory of edge-coloured multigraphs. We define the concept of regular crystallization of a manifold and prove that every non-trivial handle-free closed $n$-manifold has a regular crystallization. Then we study some applications of regular crystallizations and give a counter-example to a conjecture of Y. Tsukui [20] about strong frames of the $3$-sphere.