Abstract

For a compact orientable PL $4$-manifold $M$ with boundary, let $\hat{M}$ be the singular manifold obtained by capping of $\partial M$. In this article, we explore the class of compact orientable PL $4$-manifolds with empty or connected boundary, whose fundamental groups have rank $1$, and their corresponding singular manifolds admit weak semi-simple crystallizations. First, we show that if $M$ is a closed orientable PL $4$-manifold belonging to this class, then there exist complementary submanifolds $V$ and $V^\prime$ with a shared boundary such that $\mathcal{G}(V^\prime) \geq \mathcal{G}(M) \geq \mathcal{G}(V)$, where $\mathcal{G}(M)$, $\mathcal{G}(V)$, and $\mathcal{G}(V^\prime)$ denote the regular genera of $M$, $V$, and $V^\prime$, respectively. Next, we provide a handle decomposition for a compact orientable PL $4$-manifold $M$ with connected non-spherical boundary from this class. If the rank of the fundamental group of $\hat{M}$, $m^\prime$, is $1$, then $M$ admits a handle decomposition that takes one of the following forms:
1) one $0$-handle, one $1$-handle, $k$ $2$-handles and one $3$-handle, where $k=2+\beta_2(M)-\beta_1(M)-\beta_1(\hat{M})$, or
2) one $0$-handle, two $1$-handles, $k$ $2$-handles and one $3$-handle, where $k=3+\beta_2(M)-\beta_1(M)-\beta_1(\hat{M})$.
Further, if $m^\prime=0$, then $M$ admits a handle decomposition which consists of one $0$-handle, one $1$-handle and $\beta_2(M)$ $2$-handles. We further demonstrate that no manifold from this class with empty or connected spherical boundary can have a fundamental group isomorphic to a finite cyclic group. Finally, we provide a handle decomposition for such manifolds with empty or connected spherical boundary.

Keywords: PL-manifolds; crystallizations; regular genus; handle decomposition.

MSC: 54B15, 54C25, 57Q15, 57M15, 05C15

DOI: 10.57016/MV-eNdW6033

Pages:  1--14