Sparse handlebody decompositions and non-finiteness of $g_3=0$

TL;DR

This paper establishes a characterization of handle decompositions in PL manifolds via $k$-stacked triangulations and demonstrates the existence of infinitely many homology-spheres with $g_3=0$ in dimensions higher than four.

Contribution

It provides a new criterion linking handle decompositions to $k$-stacked triangulations and solves a longstanding problem about the non-finiteness of certain homology-spheres.

Findings

Handle decompositions correspond to $k$-stacked triangulations.

In dimensions >4, infinitely many homology-spheres have $g_3=0$.

The result answers a question posed by Kalai in 2008.

Abstract

We prove that a PL manifold admits a handle decomposition into handles of index $\le k$ if and only if $M$ is $k$-stacked, i.e., it admits a PL triangulation in which all $(d-k-1)$-faces are on $\partial M$. We use this to solve a problem posed in 2008 by Kalai: In any dimension higher than four, there are infinitely many homology-spheres with $g_3 =0$.