Representing manifolds by crystallization theory: foundations, improvements and related results

Crystallization theory was born in Italy during the 70's, due to Mario Pezzana and his school, as a combinatorial representation tool for piecewise-linear (PL) manifolds of arbitrary dimension. The present paper performes a - not exaustive - survey of the main results of PL-topology achieved through crystallization theory, both by the italian school of M. Pezzana and by other researchers of different schools, which contributed to the development of the ideas. In many cases, research problems and open questions are reviewed, together with the most recent - sometimes unpublished - results.