Planar Lefschetz fibrations and Stein structures with distinct Ozsvath-Szabo invariants on corks

TL;DR

This paper provides counterexamples to a known theorem relating Stein structures and Ozsvath-Szabo invariants on 4-manifolds with boundary, using Mazur type corks to show the converse does not always hold.

Contribution

It introduces an infinite family of counterexamples demonstrating that non-isomorphic Spinc structures do not necessarily induce contact structures with distinct Ozsvath-Szabo invariants.

Findings

Counterexamples to the converse of the Lisca-Matic-Plamenevskaya theorem

Infinite family of corks with Stein structures and non-distinct Ozsvath-Szabo invariants

Illustration of limitations in the relationship between Stein structures and contact invariants

Abstract

Thanks to a result of Lisca and Matic and a refinement by Plamenevskaya, it is known that on a 4-manifold with boundary Stein structures with non-isomorphic Spinc structures induce contact structures with distinct Ozsvath-Szabo invariants. Here we give an infinite family of examples showing that converse of Lisca-Matic-Plamenevskaya theorem does not hold in general. Our examples arise from Mazur type corks.