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

Abstract

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 $\mathbb{C}{P}^2\times D^2$ over $D^2$. These results can be easily extended to achiral as well as simplified broken (achiral) Lefschetz fibrations over $\mathbb{C}{P}^1.$ From this, it follows that every closed, connected, orientable $4$-manifold admits a smooth (simplified broken) Lefschetz fibration embedding in $\mathbb{C}{P}^2\times \mathbb{C}{P}^1.$ 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 $\mathbb{C}{P}^2\times D^2$ over $D^2$.

This article divided into two articles with more modifications. Refer, 1.Embeddings of $4$--manifolds in $S^2\times S^4$ and $S^2\tilde{\times }{S}^4$ using bordered Lefschetz fibration, Houston Journal of Mathematics, 48(3), 2022, 691--723. 2.Embeddings of $4$--manifolds in $S^4\tilde{\times }{S}^2$, Topology Proceedings, 63 (2024) pp. 57-86

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