Stein fillings of planar open books

TL;DR

This paper establishes bounds on the Euler characteristic and signature of Stein fillings for contact manifolds supported by planar open books, and classifies fillings of certain lens spaces, advancing understanding of their topology.

Contribution

It provides new finiteness results for Stein fillings of planar open books and extends classification to some lens spaces, enriching the topology of contact manifolds.

Findings

Euler characteristic and signature are bounded for Stein fillings of planar open books.

Finiteness results are proven for contact manifolds supported by spinal open books with planar pages.

Classified fillings of some lens spaces.

Abstract

We prove that if a contact manifold $(M,\xi)$ is supported by a planar open book, then Euler characteristic and signature of any Stein filling of $(M,\xi)$ is bounded. We also prove a similar finiteness result for contact manifolds supported by spinal open books with planar pages. Moving beyond the geography of Stein fillings, we classify fillings of some lens spaces.