Handle-decompositions of PL 4-manifolds

The present paper studies the relationship between handle-decompositions of PL 4-manifolds and the so called "crystallization theory", which represents PL n-manifolds by means of pseudosimplicial triangulations admitting exactly n+1 vertices. Within this theory, the combinatorial invariant "regular genus" plays a central role. In particular, the authors obtain the characterization of PL 4-manifolds $M^4$ (with empty or connected boundary $\partial M^4$) satisfying suitable conditions concerning the regular genus $G(M^4)$, the rank of the fundamental group $rk(\pi_1(M^4))$, and the boundary regular genus $G(\partial M^4)$. As a consequence, it is completed the classification of PL 4-manifolds with (possibly disconnected) boundary up to regular genus two.