Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Homotopy theoretic properties of open books

Ruizhi Huang, Stephen Theriault

Abstract

We study the homotopy groups of open books in terms of those of their pages and bindings. Under homotopy theoretic conditions on the monodromy we prove an integral decomposition result for the based loop space on an open book, and under more relaxed conditions prove a rational loop space decomposition. The latter case allows for a rational dichotomy theorem for open books, as an extension of the classical dichotomy in rational homotopy theory. As a direct application, we show that for Milnor's open book decomposition of an odd sphere with monodromy of finite order the induced action of the monodromy on the homology groups of its page cannot be nilpotent.

Homotopy theoretic properties of open books

Abstract

We study the homotopy groups of open books in terms of those of their pages and bindings. Under homotopy theoretic conditions on the monodromy we prove an integral decomposition result for the based loop space on an open book, and under more relaxed conditions prove a rational loop space decomposition. The latter case allows for a rational dichotomy theorem for open books, as an extension of the classical dichotomy in rational homotopy theory. As a direct application, we show that for Milnor's open book decomposition of an odd sphere with monodromy of finite order the induced action of the monodromy on the homology groups of its page cannot be nilpotent.

Paper Structure

This paper contains 7 sections, 11 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. An integral loop space decomposition of open books with homotopically trivial monodromy
  3. A loop space decomposition of the double of $V$
  4. A rational loop space decomposition of certain open books
  5. The proof of Theorem \ref{['openbookthm']}
  6. The necessity of the homotopy conditions on the monodromy
  7. Mather's cube theorems

Key Result

Theorem 1.1

Let $M$ be a path-connected open book for which there is a diffeomorphism $M\cong (\partial V\times D^2)\cup_{\rm id} V_h$. Suppose that $h\simeq {\rm id}$ relative to $\partial V$. Then there is a homotopy equivalence where the space $F$ is the homotopy fibre of the inclusion $\partial V\stackrel{}{\rightarrow} V$. Consequently, there is an isomorphism

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • proof : Proof of Theorem \ref{['productpushout']}
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem \ref{['zMdecthm']}
  • Lemma 3.1
  • proof
  • ...and 14 more