Lower bounds for regular genus and gem-complexity of PL 4-manifolds

Within crystallization theory, two interesting PL invariants for $d$-manifolds have been introduced and studied, namely {\it gem-complexity} and {\it regular genus}. In the present paper we prove that, for any closed connected PL $4$-manifold $M$, its gem-complexity $k(M)$ and its regular genus $\mathcal{G}(M)$ satisfy: $$k(M)\ge 3\chi (M)+10m-6\text{ and }\mathcal{G}(M)\ge 2\chi (M)+5m-4,$$ where $rk({\pi }_1(M))=m\mathrm{.}$ These lower bounds enable to strictly improve previously known estimations for regular genus and gem-complexity of product 4-manifolds. Moreover, the class of {\it semi-simple crystallizations} is introduced, so that the represented PL 4-manifolds attain the above lower bounds. The additivity of both gem-complexity and regular genus with respect to connected sum is also proved for such a class of PL 4-manifolds, which comprehends all ones of “standard type”, involved in existing crystallization catalogues, and their connected sums.