Research Projects

A crystallization of a PL d-manifold (possibly with boundary) is a certain type of edge colored graph which represents the manifold. The journey of crystallization theory has begun due to Pezzana who gives the existence of a crystallization for every closed connected PL d-manifold, and later for every PL d-manifold with boundary. Extending the notion of genus in dimension 2, the notion of regular genus G(M) for a d-manifold M has been introduced which is strictly related to the existence of regular embeddings of crystallizations of the manifold into surfaces. Later, the concept of regular genus has been extended for a d-manifold with boundary, for d greater than 1. Given a PL d-manifold M, its gem-complexity k(M) is the non-negative integer p, where 2p is the minimum number of vertices of a crystallization of M. This project contains several problems related to regular genus and gem-complexity of PL d-manifolds (possibly with boundary).