On regular genus and G-degree of PL 4-manifolds with boundary

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: $$\tilde{G} (M) \geq 2 \chi(M)+3m+2h-4+2 \hat{m} \mbox{ and } \tilde{D}_G (M) \geq 12(2 \chi(M)+3m+2h-4+2 \hat{m}),$$ 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: $$\mathcal{G} (M) \geq 2 \chi(M)+3m+2h-4+2 \hat{m},$$ 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.