Let π be a connected compact PL 4-manifold with boundary.
In this article, we give several lower bounds for regular genus and gem-complexity of the manifold π.
In particular, we prove that if π is a connected compact 4-manifold with β boundary components, then its gem-complexity k(M) satisfies the inequalities k(M)\geq 3\chi(M)+7m+7h-10 and k(M)\geq k(\partial M)+3\chi(M)+4m+6h-9, and its regular genus \mathcal{G}(M) satisfies the inequalities \mathcal{G}(M)\geq 2\chi(M)+3m+2h-4 and \mathcal{G}(M)\geq\mathcal{G}(\partial M)+2\chi(M)+2m+2h-4, where π is the rank of the fundamental group of the manifold π.
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 is also shown for a large class of PL 4-manifolds with boundary.
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