(Achiral) Lefschetz fibration embeddings of $4$-manifolds

In this paper, we prove Lefschetz fibration embeddings of achiral as well as simplified broken (achiral) Lefschetz fibrations of compact, connected, orientable $4$-manifolds over $D^2$ into the trivial Lefschetz fibration of $\mathbb CP^2\times D^2$ over $D^2$. These results can be easily extended to achiral as well as simplified broken (achiral) Lefschetz fibrations over $\mathbb CP^1.$ From this, it follows that every closed, connected, orientable $4$-manifold admits a smooth (simplified broken) Lefschetz fibration embedding in $\mathbb CP^2\times \mathbb CP^1.$ We provide a huge collection of bordered Lefschetz fibration which admit bordered Lefschetz fibration embeddings into a trivial Lefschetz fibration $\tilde\pi:D^4\times D^2\to D^2.$ We also show that every closed, connected, orientable $4$-manifold $X$ admits a smooth embedding into $S^4\times S^2$ as well as into $S^4\tilde\times S^2$. From this, we get another proof of a theorem of Hirsch which states that every closed, connected, orientable $4$-manifold smoothly embeds in $\mathbb R^7.$ We also discuss Lefschetz fibration embedding of non-orientable $4$-manifolds $X$, where $X$ does not admit $3$- and $4$-handles in the handle decomposition, into the trivial Lefschetz fibration of $\mathbb CP^2\times D^2$ over $D^2$.