Compact 4-manifolds admitting special handle decompositions

TL;DR

This paper investigates special handle decompositions of compact PL 4-manifolds, identifying classes that admit such decompositions and can be represented by framed links, especially those minimizing certain combinatorial invariants.

Contribution

It introduces a class of simply-connected PL 4-manifolds with special handle decompositions, linking them to minimized combinatorial invariants and framed link representations.

Findings

Identifies classes of 4-manifolds with special handle decompositions.

Shows these manifolds can be represented by undotted framed links.

Connects handle decompositions to minimized combinatorial invariants.

Abstract

In this paper we study colored triangulations of compact PL 4-manifolds with empty or connected boundary which induce handle decompositions lacking in 1-handles or in 1- and 3-handles, thus facing also the problem, posed by Kirby, of the existence of {\it special handlebody decompositions} for any simply-connected closed PL 4-manifold. In particular, we detect a class of compact simply-connected PL 4-manifolds with empty or connected boundary, which admit such decompositions and, therefore, can be represented by (undotted) framed links. Moreover, this class includes any compact simply-connected PL 4-manifold with empty or connected boundary having colored triangulations that minimize the combinatorially defined PL invariant {\em regular genus, gem-complexity} or {\em G-degree} among all such manifolds with the same second Betti number.