论文标题
PL 4-manifolds的常规属和G度
On regular genus and G-degree of PL 4-manifolds with boundary
论文作者
论文摘要
在本文中，我们介绍了两个新的PL-Invarianiant：加权常规属和具有边界的歧管加权G-Degree。我们首先证明了两种不平等现象，涉及一些不变性，这些不平等表明，对于任何带有非球形边界组件的PL-manifold $ m $，$ m $的常规属$ \ natus $ \ natus $ \ natus $ \ natercal {g}（m）$至少是加权的常规$ \ tilde $ \ tilde {g}（g}（m）$ m $，至少是$ g $ $ g $ g $ $ $ $ $ $ $。 $ m $。另一种不等式指出，加权g-degrey $ \ tilde {d} _g（m）$ m $始终大于或等于$ m $的g-begree $ d_g（m）$。令$ m $为任何紧凑的连接pl $ 4 $ - manifold，$ h $的非球形边界组件数量。然后，我们计算以下内容：$ \ tilde {g}（m）\ geq2χ（m）+3m+2H-4+2 \ 2 \ hat {m} \ mbox {and} \ tilde {d} _g（d} _g（m） $ \ hat {m} $是$ m $的基本组和相应的奇异歧管$ \ wideHat {m} $的等级（通过分别从$ m $的边界组件中获得。作为 结果我们证明了常规属$ \ Mathcal {g}（m）$满足以下不平等： $$ \ MATHCAL {G}（M）\ GEQ2χ（M）+3M+3M+2H-4+2 \ HAT {M}，$$ 它改善了$ m $的常规属$ \ mathcal {g}（m）$的已知下限。然后，我们定义两类的宝石，$ 4 $ - manifold $ m $带边界：一种由半简单的宝石组成，另一个由弱的半简单宝石组成，并证明，加权G度和加权的常规属的下限分别在这两个类别中获得。
In this article, we introduce two new PL-invariants: weighted regular genus and weighted G-degree for manifolds with boundary. We first prove two inequalities involving some PL-invariants which state that for any PL-manifold $M$ with non spherical boundary components, the regular genus $\mathcal{G}(M)$ of $M$ is at least the weighted regular genus $\tilde{G}(M)$ of $M$ which is again at least the generalized regular genus $\bar{G}(M)$ of $M$. Another inequality states that the weighted G-degree $\tilde{D}_G (M)$ of $M$ is always greater than or equal to the G-degree $D_G (M)$ of $M$. Let $M$ be any compact connected PL $4$-manifold with $h$ number of non spherical boundary components. Then we compute the following: $$\tilde{G} (M) \geq 2 χ(M)+3m+2h-4+2 \hat{m} \mbox{ and } \tilde{D}_G (M) \geq 12(2 χ(M)+3m+2h-4+2 \hat{m}),$$ where $m$ and $\hat{m}$ are the ranks of the fundamental groups of $M$ and the corresponding singular manifold $\widehat{M}$ (obtained by coning off the boundary components of $M$) respectively. As a consequence we prove that the regular genus $\mathcal{G}(M)$ satisfies the following inequality: $$\mathcal{G} (M) \geq 2 χ(M)+3m+2h-4+2 \hat{m},$$ which improves the previous known lower bounds for the regular genus $\mathcal{G}(M)$ of $M$. Then we define two classes of gems for PL $4$-manifold $M$ with boundary: one consists of semi-simple gems and the other consists of weak semi-simple gems, and prove that the lower bounds for the weighted G-degree and weighted regular genus are attained in these two classes respectively.
