Let $M$ be a compact connected 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 showed for a large class of PL $4$-manifolds with boundary.

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具有边界的 PL 4-流形的正则属和宝石复杂度的下限

令 $M$ 是具有边界的紧凑连通 PL 4-流形。在本文中，我们给出了流形 $M$ 的正则属和 gem-complexity 的几个下界。特别地，我们已经证明，如果 $M$ 是具有 $h$ 边界分量的连通紧致 $4$-流形，那么它的 gem-complexity $\mathit{k}(M)$ 满足以下不等式： $$\mathit{ k}(M)\geq 3\chi(M)+7m+7h-10 \mbox{ 和 }\mathit{k}(M)\geq \mathit{k}(\partial M)+3\chi(M )+4m+6h-9,$$ 及其正则属 $\mathcal{G}(M)$ 满足以下不等式： $$\mathcal{G}(M)\geq 2\chi(M)+3m+ 2h-4\mbox{ 和 }\mathcal{G}(M)\geq \mathcal{G}(\partial M)+2\chi(M)+2m+2h-4,$$ 其中 $m$ 是流形 $M$ 的基本群的秩。这些下限能够严格改进先前已知的对具有边界的 PL $4$-流形的常规属和宝石复杂度的估计。此外，对于一大类具有边界的 PL$4$-流形也显示了这些边界的锐度。