Handle decompositions for a class of closed orientable PL 4-manifolds

TL;DR

This paper constructs specific handle decompositions for a class of closed orientable PL 4-manifolds with infinite cyclic fundamental group, linking the number of 2-handles to the second Betti number.

Contribution

It provides explicit handle decompositions for certain PL 4-manifolds with semi-simple crystallizations and infinite cyclic fundamental group, detailing the handle counts based on topological invariants.

Findings

Handle decompositions depend on the second Betti number.

Number of 2-handles varies with the Betti number.

At most 2 handles of certain types are needed.

Abstract

In this article, we study a class of closed connected orientable PL $4$-manifolds admitting a semi-simple crystallization and which have an infinite cyclic fundamental group. We show that the manifold in the class admits a handle decomposition in which the number of $2$-handles depends upon its second Betti number and other $h$-handles ($h \leq 4$) are at most $2$. More precisely, our main result is the following. For a closed connected orientable PL $4$-manifold having a semi-simple crystallization with the fundamental group as $\mathbb{{Z}}$, we have constructed a handle decomposition for $M$ as one of the following types: $(1)$ one $0$-handle, two $1$-handles, $1+\beta_2(M)$ $2$-handles, one $3$-handle and one $4$-handle, $(2)$ one $0$-handle, one $1$-handle, $\beta_2(M)$ $2$-handles, one $3$-handle and one $4$-handle, where $\beta_2(M)$ denotes the second Betti number of manifold $M$ with $\mathbb{Z}$ coefficients.