论文标题
定期属的下限和PL 4 manifolds的宝石复杂性具有边界
Lower bounds for regular genus and gem-complexity of PL 4-manifolds with boundary
论文作者
论文摘要
令$ m $为带边界的连接的紧凑型PL 4-manifold。在本文中，我们为歧管$ m $的常规属和宝石复杂性提供了几个下限。特别是，我们已经证明，如果$ m $是连接的紧凑型$ 4 $ - manifold，$ h $边界组件，那么其宝石 - 复杂性$ \ mathit {k}（k}（m）$都满足以下不平等： $$ \ MATHIT {K}（M）\ GEQ3χ（M）+7M+7H-10 \ Mbox {and} \ Mathit {k}（M）\ GEQ \ GEQ \ MATHIT {K}（k}（\ partial m） 及其常规属$ \ Mathcal {g}（m）$满足以下不平等： $$ \ MATHCAL {G}（M）\ GEQ2χ（M）+3M+2H-4 \ Mbox {and} \ Mathcal {G}（M）\ GEQ \ GEQ \ MATHCAL {G} 其中$ m $是歧管$ m $的基本组的排名。这些下限可以严格改善对带有边界的PL $ 4 $ manifold的常规属和宝石复杂性的先前已知估计。此外，这些边界的清晰度也已显示出一大类带边界的PL $ 4 $ manifolds。
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: $$\mathit{k}(M)\geq 3χ(M)+7m+7h-10 \mbox{ and }\mathit{k}(M)\geq \mathit{k}(\partial M)+3χ(M)+4m+6h-9,$$ and its regular genus $\mathcal{G}(M)$ satisfies the following inequalities: $$\mathcal{G}(M)\geq 2χ(M)+3m+2h-4\mbox{ and }\mathcal{G}(M)\geq \mathcal{G}(\partial M)+2χ(M)+2m+2h-4,$$ 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.
