论文标题
（Achiral）Lefschetz纤维化嵌入$ 4 $ -Manifolds
(Achiral) Lefschetz fibration embeddings of $4$-manifolds
论文作者
论文摘要
在本文中，我们证明了Achiral的Lefschetz纤维化嵌入以及简化的破碎（Acthiral）Lefschetz纤维纤维的纤维纤维纤维纤维纤维纤维纤维，$ 4 $ -Manifolds $ d^2 $超过$ d^2 $ to the Trivial Lefschetz纤维纤维的$ \ \ Mathbb cp cp cp^2 $ $ \ \ \ \ \ \ \ \ \ mathbb cp^2 $。这些结果可以很容易地扩展到Achiral以及简化的破碎（ACHIRAL）LEFSCHETZ纤维，超过$ \ Mathbb cp^1. $ $，可以遵循每个封闭的，连接的，可定向的$ 4 $ -MANIFOLD承认平稳的（简化的破碎）Lefschetz纤维固定在$ \ Mathbb cp^2. Mathbb cp^2. $ cp^1. $ \ cp cp^1. cp^2. 1.接壤的lefschetz振动，接收到将lefschetz纤维嵌入到琐碎的lefschetz纤维中$ \tildeπ：d^4 \ times d^2 \ to d^2. $我们还显示，我们还显示每个封闭，连接，可定位，定向$ 4 $ -manifold $ x $ s $ s $ s $ s^4 $ s^4 $ s^4 $ 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 $ \ mathbb cp^2 \ times d^2 $ of $ d^2 $的Lefschetz纤维化。
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 $\tildeπ:D^4\times D^2\to D^2.$ 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 $D^2$.
