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

Karim Adiprasito, Bruno Benedetti

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 $g_3 =0$.

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

Abstract

We prove that a PL manifold admits a handle decomposition into handles of index if and only if is -stacked, i.e., it admits a PL triangulation in which all -faces are on . 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 .

Paper Structure

This paper contains 2 sections, 4 theorems, 1 figure.

Table of Contents

  1. Proof of Theorem \ref{['thm:handlebody']}
  2. Proof of Theorem \ref{['thm:B']}

Key Result

Lemma 1

Let $1 \le k \le d$ be integers. Let $C$ be any $(k-1)$-complex PL-embedded inside a $(d-1)$-complex $N$. Let $T$ be a PL triangulation of $N$ such that $C$ is transversal to $T$. There is a stellar subdivision $T'$ of $T$ such that

Figures (1)

  • Figure 1: To subdivide stellarly the boundary of some $3$-manifold, at a triangle (in pink) and at an edge (in red), we stack at the pink triangle, and we build a pyramid (consisting of two tetrahedra) on top of the star of the red edge. The boundary of the new $3$-manifold is the stellar subdivision we desired. The pink triangle, the red edge, and the two triangles containing the red edge, are now interior. No vertex of the original $3$-manifold has become interior.

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • proof
  • Lemma 5
  • proof
  • proof : Proof of Theorem B