Crystallization theory is a graph-theoretical representation method for compact PL-manifolds of arbitrary dimension, which makes use of a particular class of edge-coloured graphs, which are dual to coloured (pseudo-)triangulations. The purely combinatorial nature of crystallizations makes them particularly suitable for automatic generation and classication, as well as for the introduction and study of graph-defined invariants for PL-manifolds. The present survey paper focuses on the 4-dimensional case, presenting up-to-date results about the PL classication of closed 4-manifolds, by means of two such PL invariants: regular genus and gem-complexity. Open problems are also presented, mainly concerning different classication of 4-manifolds in TOP and DIFF=PL categories, and a possible approach to the 4-dimensional Smooth Poincare Conjecture.

Classifying PL 4-manifolds via crystallizations: results and open problems / Casali, Maria Rita; Cristofori, Paola; Gagliardi, Carlo. - ELETTRONICO. - (2016), pp. 199-226.

Classifying PL 4-manifolds via crystallizations: results and open problems

CASALI, Maria Rita;CRISTOFORI, Paola;GAGLIARDI, Carlo
2016

Abstract

Crystallization theory is a graph-theoretical representation method for compact PL-manifolds of arbitrary dimension, which makes use of a particular class of edge-coloured graphs, which are dual to coloured (pseudo-)triangulations. The purely combinatorial nature of crystallizations makes them particularly suitable for automatic generation and classication, as well as for the introduction and study of graph-defined invariants for PL-manifolds. The present survey paper focuses on the 4-dimensional case, presenting up-to-date results about the PL classication of closed 4-manifolds, by means of two such PL invariants: regular genus and gem-complexity. Open problems are also presented, mainly concerning different classication of 4-manifolds in TOP and DIFF=PL categories, and a possible approach to the 4-dimensional Smooth Poincare Conjecture.
A MATHEMATICAL TRIBUTE to PROFESSOR JOSE MARIA MONTESINOS AMILIBIA
CASTRILLON, M.; MARTIN-PEINADOR, E.; RODRIGUES-SANJURJO, J.M.; RUIZ, J.M.
9788460816843
ULZAMA DIGITAL
SPAGNA
Classifying PL 4-manifolds via crystallizations: results and open problems / Casali, Maria Rita; Cristofori, Paola; Gagliardi, Carlo. - ELETTRONICO. - (2016), pp. 199-226.
Casali, Maria Rita; Cristofori, Paola; Gagliardi, Carlo
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11380/1108852
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