2020/12/23 by Pandit, Suhas, A, Selvakumar
#57R15 #57R40 #57R45 #57R65 #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2012.12644
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 D2 into the trivial Lefschetz fibration of \mathbb CP2× D2 over D2. These results can be easily extended to achiral as well as simplified broken (achiral) Lefschetz fibrations over \mathbb CP1. From this, it follows that every closed, connected, orientable 4-manifold admits a smooth (simplified broken) Lefschetz fibration embedding in \mathbb CP2× \mathbb CP1. We provide a huge collection of bordered Lefschetz fibration which admit bordered Lefschetz fibration embeddings into a trivial Lefschetz fibration π:D4× D2→ D2. We also show that every closed, connected, orientable 4-manifold X admits a smooth embedding into S4× S2 as well as into S4× S2. From this, we get another proof of a theorem of Hirsch which states that every closed, connected, orientable 4-manifold smoothly embeds in \mathbb R7. 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 CP2× D2 over D2.