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Handle decompositions and stabilizations of open books

Chun-Sheng Hsueh

TL;DR

The paper develops a unified handle-theoretic framework to study open book decompositions on $n$-manifolds ($n\ge 3$), introducing symmetric handle decompositions and exchange moves that transform pages while preserving the ambient manifold. It establishes $k$-stabilizations and middle-dimensional stabilizations, showing how they correspond to Hopf plumbing in low dimensions and generalize prior stabilization notions. A key application demonstrates that open books with trivial monodromy can be stabilized so that their pages become boundary connected sums of trivial disk bundles over spheres, and it proves a common-page stabilization principle for pairs of open books with matching Euler characteristics (with parity caveats). The results unify high-dimensional stabilization phenomena, clarify the relationship with classical plumbings, and provide explicit constructions that facilitate the manipulation and classification of open books in dimensions $n\ge 3$.

Abstract

We build handle decompositions of n-manifolds that encode given open book decompositions and describe handle slides that reveal new open book decompositions on the same underlying manifold, for $n \geq 3$. This recovers known stabilization operations for open books. As an application, we show that any open book with trivial monodromy can be stabilized to an open book whose page is a boundary connected sum of trivial disk bundles over spheres.

Handle decompositions and stabilizations of open books

TL;DR

The paper develops a unified handle-theoretic framework to study open book decompositions on -manifolds (), introducing symmetric handle decompositions and exchange moves that transform pages while preserving the ambient manifold. It establishes -stabilizations and middle-dimensional stabilizations, showing how they correspond to Hopf plumbing in low dimensions and generalize prior stabilization notions. A key application demonstrates that open books with trivial monodromy can be stabilized so that their pages become boundary connected sums of trivial disk bundles over spheres, and it proves a common-page stabilization principle for pairs of open books with matching Euler characteristics (with parity caveats). The results unify high-dimensional stabilization phenomena, clarify the relationship with classical plumbings, and provide explicit constructions that facilitate the manipulation and classification of open books in dimensions .

Abstract

We build handle decompositions of n-manifolds that encode given open book decompositions and describe handle slides that reveal new open book decompositions on the same underlying manifold, for . This recovers known stabilization operations for open books. As an application, we show that any open book with trivial monodromy can be stabilized to an open book whose page is a boundary connected sum of trivial disk bundles over spheres.

Paper Structure

This paper contains 10 sections, 12 theorems, 53 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Application
  4. Preliminaries
  5. Handle decompositions
  6. Half open books
  7. A symmetric handle decomposition on open books
  8. Handle exchange
  9. Main results and relation to previously known stabilizations
  10. Application

Key Result

Theorem 1.1

Let $(M,\varphi)$ be an open book decomposition on $X$ of dimension $n\geq 3$. Then for each integer $k\in[2,n-1]$, there exists a monodromy map $\varphi_k$ on $M \mathop{\natural} (S^{k-1}\times D^{n-k})\mathop{\natural} (S^{n-k}\times D^{k-1})$ such that $\varphi_k$ restricts to $\varphi$ on $M$, is an open book decomposition on $X$ not equivalent to $(M,\varphi)$.

Figures (11)

  • Figure 1: Constructing the half open book with page $[0,1]$.
  • Figure 2: The solid torus is obtained from the annulus times half-interval by collapsing $\{p\}\times[0,\frac{1}{2}]$ to a point for all points $p$ in the boundary of the annulus.
  • Figure 3: Trivial monodromy case: $a({\mathbf{h}^1}^{*})$ and $b(\mathbf{h}^1)$ are isotopic to $coc(\operatorname{h}^1)\cup coc(\operatorname{h}^1)\subset \partial\operatorname{hob}(M)\cong M\times\{0\}\cup_{\partial M}M\times\{\frac{1}{2}\}$.
  • Figure 4:
  • Figure 5: Isotoping the attaching sphere $a({\mathbf{h}_j^k}^*)$ of the dual of a selected handle $\mathbf{h}_j^k$ into the boundary of the $0$-handle.
  • ...and 6 more figures

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof : Proof outline
  • Definition 2.4
  • ...and 30 more