Embeddings of 4-manifolds in $S^4 \tilde{×} S^2$ - Pandit, Suhas; A, Selvakumar (Topology Proceedings, 2023)
In this paper, we discuss simplified broken achiral Lefschetz fibration embeddings (SBLFE) of simplified broken achiral Lefschetz fibrations (SBLF) of compact, connected, oriented, smooth 4 -manifolds over $D^2$ into the trivial Lefschetz fibration of $\mathcal{D} E(1) × D^2$ over $D^2$. Using this, we show that the double of an SBLF admits a smooth embedding in $S^4 \overline{×} S^2$. Also, we provide broken achiral Lefschetz fibration (BLF) embeddings of a huge collection of BLF's over $S^2$ into the trivial Lefschetz fibration of $S^4 × S^2$ over $S^2$. Finally, given a closed oriented smooth 4 -manifold $Y$, we provide BLF embeddings of a huge collection of BLF's over $S^2$ into the trivial Lefschetz fibration of $Y × S^2$ over $S^2$.