arXiv 2005.107722020-05math.GT

光滑切触流形的开书与嵌入

On open books and embedding of smooth and contact manifolds

Arijit Nath, Kuldeep Saha

备注：: 19 pages, 6 figures

摘要：

讨论了流形在开书、切触流形和切触开书范畴中的嵌入。我们证明了Haefliger-Hirsch嵌入定理的一个开卷版本，证明了每个$k$ -连通的闭$n$ -流形（$n\geq 7$，$k < \frac{n-4}{2}$）在$\mathbb{S}^{2n-k}$的平凡开卷中有一个开卷嵌入.然后，我们证明了每一个封闭的流形$M^{2n+1}$的边界的非手性Lefschetz纤维化，承认开书嵌入在平凡的开书$\mathbb{S}^{2\lfloor\frac{3n}{2}\rfloor + 3}$。证明了每一个包围非手性Lefschetz纤维化的闭流形$M^{2n+1}$都有一个切触结构，该切触结构等切触嵌入到$\mathbb{R}^{2n+3}.$上的标准切触结构中。最后，给出了切触$(2n+1)$ -流形的切触开书嵌入到$\mathbb{S}^{4n+1}.$上的标准切触结构的平凡支撑开书中的各种例子

英文摘要：

We discuss embedding of manifolds in the category of open books, contact manifolds and contact open books. We prove an open book version of the Haefliger--Hirsch embedding theorem by showing that every $k$-connected closed $n$-manifold ($n\geq 7$, $k < \frac{n-4}{2}$) admits an open book embedding in the trivial open book of $\mathbb{S}^{2n-k}$. We then prove that every closed manifold $M^{2n+1}$ that bounds an achiral Lefschetz fibration, admits open book embedding in the trivial open book of $\mathbb{S}^{2\lfloor\frac{3n}{2}\rfloor + 3}$. We also prove that every closed manifold $M^{2n+1}$ bounding an achiral Lefschetz fibration admits a contact structure that isocontact embeds in the standard contact structure on $\mathbb{R}^{2n+3}.$ Finally, we give various examples of contact open book embeddings of contact $(2n+1)$-manifolds in the trivial supporting open book of the standard contact structure on $\mathbb{S}^{4n+1}.$

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