Open books for Boothby-Wang bundles, fibered Dehn twists and the mean Euler characteristic

TL;DR

This paper studies contact manifolds arising from open books with fibered Dehn twists, using equivariant symplectic homology to distinguish them and show certain twists are not symplectically trivial.

Contribution

It introduces a method to distinguish contact manifolds via mean Euler characteristic, demonstrating that some fibered Dehn twists are not symplectically isotopic to the identity.

Findings

Fibered Dehn twists can produce distinct contact manifolds.

Mean Euler characteristic distinguishes non-isotopic twists.

Some fibered Dehn twists are not symplectically trivial.

Abstract

We examine open books with powers of fibered Dehn twists as monodromy. The resulting contact manifolds can be thought of as Boothby-Wang orbibundles over symplectic orbifolds. Using the mean Euler characteristic of equivariant symplectic homology we can distinguish these contact manifolds and hence show that some fibered Dehn twists are not symplectically isotopic to the identity relative to the boundary. This complements results of Biran and Giroux.