A note on embedding of achiral Lefschetz fibrations

TL;DR

This paper explores embeddings of achiral Lefschetz fibrations in higher-dimensional manifolds, providing new proofs and conditions for embedding in specific 6-dimensional spaces, with implications for 4-manifold theory.

Contribution

It offers a new proof that all closed orientable 4-manifolds embed in S^2×S^2×S^2 and characterizes conditions for embedding achiral Lefschetz fibrations with hyperelliptic monodromy in D^6.

Findings

Every closed orientable 4-manifold embeds in S^2×S^2×S^2.

Achiral Lefschetz fibrations with hyperelliptic monodromy embed in D^6.

An obstruction to LF embeddings is discussed.

Abstract

We discuss $4$-dimensional achiral Lefschetz fibrations bounding $3$-dimensional open books and study their Lefschetz fibration (LF) embedding in a bounded $6$-dimensional manifold, in the sense of Ghanwat--Pancholi. As an application we give another proof of the fact that every closed orientable $4$-manifold embeds in $S^2 \times S^2 \times S^2$ . We also show that every achiral Lefschetz fibration with hyperelliptic monodromy admits LF embedding in $D^6 = D^2 \times D^4$ and discuss an obstruction to such LF embeddings.