Homotopy theoretic properties of open books

TL;DR

This paper investigates the homotopy groups of open books, providing decomposition results under certain conditions and applying these findings to Milnor's open book decomposition of spheres, revealing new insights into their homotopy properties.

Contribution

It offers new homotopy decomposition theorems for open books based on monodromy conditions and extends classical rational homotopy dichotomy to this context.

Findings

Integral decomposition of based loop space under homotopy conditions

Rational loop space decomposition with relaxed conditions

Monodromy action on homology cannot be nilpotent for Milnor's open book of spheres

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.