  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.

Geometric Topology;Combinatorics