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A note on codimension $2$ spun embedding

Sneha Banerjee, Shital Lawande, Subhadeep Rana, Kuldeep Saha

TL;DR

The paper proves that if a binding component $B$ of an open book on $M$ carries an open book, then every open book on $B$ spun embeds in $M$, via a generalized page-trick built on Seifert-type embeddings and ambient isotopies realizing monodromies. This yields far-reaching codimension-2 spun embeddings, including that any open book on a simply connected spin $5$-manifold spun embeds in $S^7$ and any $3$-dimensional open book spun embeds in $S^5$, and it introduces Morse open books and a Morse version of the page trick. The method hinges on a generalization of the Hopf annulus trick to a broad “page trick” framework, enabling monodromy powers to be realized as ambient diffeomorphisms and enabling concatenations that produce branched-cover total spaces. The results connect to Brieskorn manifolds and generalized plumbing, and extend to Morse open books, widening the scope of feasible spun embeddings in high-dimensional spheres with potential implications for contact-topological embeddings and universal open-book questions.

Abstract

We prove that if a closed manifold $B$ is a connected component of the binding of an open book decomposition of a manifold $M$, then every open book decomposition of $B$ spun embeds in $M$. As an application, we prove that every open book decomposition of a simply connected spin $5$-manifold spun embeds in $S^7$ and every $3$-dimensional open book spun embeds in $S^5$. We also define a notion of spun embedding for Morse open books.

A note on codimension $2$ spun embedding

TL;DR

The paper proves that if a binding component of an open book on carries an open book, then every open book on spun embeds in , via a generalized page-trick built on Seifert-type embeddings and ambient isotopies realizing monodromies. This yields far-reaching codimension-2 spun embeddings, including that any open book on a simply connected spin -manifold spun embeds in and any -dimensional open book spun embeds in , and it introduces Morse open books and a Morse version of the page trick. The method hinges on a generalization of the Hopf annulus trick to a broad “page trick” framework, enabling monodromy powers to be realized as ambient diffeomorphisms and enabling concatenations that produce branched-cover total spaces. The results connect to Brieskorn manifolds and generalized plumbing, and extend to Morse open books, widening the scope of feasible spun embeddings in high-dimensional spheres with potential implications for contact-topological embeddings and universal open-book questions.

Abstract

We prove that if a closed manifold is a connected component of the binding of an open book decomposition of a manifold , then every open book decomposition of spun embeds in . As an application, we prove that every open book decomposition of a simply connected spin -manifold spun embeds in and every -dimensional open book spun embeds in . We also define a notion of spun embedding for Morse open books.

Paper Structure

This paper contains 10 sections, 8 theorems, 1 equation, 3 figures.

Table of Contents

  1. Introduction
  2. preliminaries
  3. Open books and embeddings
  4. Seifert type embedding
  5. Isotopy via the flow of an open book
  6. The annulus trick
  7. A generalization of the annulus trick
  8. Proof of the main theorem
  9. The page trick in a more general setup
  10. The page trick for Morse open books

Key Result

Theorem 1.1

Let $M$ be a closed orientable $n$-manifold with an open book decomposition $\textrm{OB}(P^{n-1},\rho)$. Let $B^{n-2}$ be a connceted component of the binding $\partial P$ and let $\textrm{OB}(V_B,\phi_B)$ be an open book decomposition of $B$. Then, $\textrm{OB}(V_B, \phi_B^k)$ spun embeds in $\text

Figures (3)

  • Figure 1: Isotoping $\Sigma \subset M \times \{0\}$ to a Seifert type embedding in $M \times [0,1]$.
  • Figure 2: Hopf annuli
  • Figure 3: The purple and green lines represent $g_1(P_1)$ and $g_2(P_2)$, respectively, in $M \times [-2,1]$ and $s$ is a coordinate on the second factor of $\Sigma \times [0,1]$. We first isotope $g_1(P_1)$ to induce $\rho_1$ at $s = \frac{1}{3}$, and then isotope $g_2(P_2)$ to induce $\rho_2$ at $s =\frac{2}{3}$. This induces the monodromy $\rho_1 \circ \rho_2$ on $V$.

Theorems & Definitions (16)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Definition 2.1: Spun embedding
  • Proposition 2.2: The Hopf annulus trick hy,pps
  • Proposition 2.3: The page trick
  • proof : Proof of Theorem \ref{['thm0']}
  • Lemma 4.1
  • Example 4.1
  • ...and 6 more