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

Let $M$ be a connected compact PL 4-manifold with boundary. In this article, we have given several lower bounds for regular genus and gem-complexity of the manifold $M$. In particular, we have proved that if $M$ is a connected compact $4$-manifold with $h$ boundary components then its gem-complexity $\mathit{k}(M)$ satisfies the following inequalities: $$\mathit{k}(M)\geq 3\chi(M)+7m+7h-10 \mbox{ and }\mathit{k}(M)\geq \mathit{k}(\partial M)+3\chi(M)+4m+6h-9,$$ and its regular genus $\mathcal{G}(M)$ satisfies the following inequalities: $$\mathcal{G}(M)\geq 2\chi(M)+3m+2h-4\mbox{ and }\mathcal{G}(M)\geq \mathcal{G}(\partial M)+2\chi(M)+2m+2h-4,$$ where $m$ is the rank of the fundamental group of the manifold $M$. These lower bounds enable to strictly improve previously known estimations for regular genus and gem-complexity of a PL $4$-manifold with boundary. Further, the sharpness of these bounds has also been shown for a large class of PL $4$-manifolds with boundary.