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 $k(M)$ satisfies the following inequalities: $$k(M)\ge 3\chi (M)+7m+7h-10andk(M)\ge k(\mathrm{\partial }M)+3\chi (M)+4m+6h-9,$$ and its regular genus $\mathcal{G}(M)$ satisfies the following inequalities: $$\mathcal{G}(M)\ge 2\chi (M)+3m+2h-4and\mathcal{G}(M)\ge \mathcal{G}(\mathrm{\partial }M)+2\chi (M)+2m+2h-4,$$ 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.