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

TL;DR

This paper introduces new PL-invariants for 4-manifolds with boundary, establishes inequalities relating these invariants to topological features, and computes lower bounds that improve previous estimates, also defining special classes of gems.

Contribution

The paper defines weighted regular genus and G-degree for PL 4-manifolds with boundary, derives new inequalities, and identifies classes of gems where bounds are attained, advancing the understanding of manifold invariants.

Findings

New inequalities relating PL-invariants and topological features.

Computed lower bounds for regular genus and G-degree that improve previous results.

Identified classes of gems where bounds are sharp.

Abstract

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.