These days, I have been doing a small project in terms of sub-Riemannian limits on contact manifolds. I came to learn that there was a profound relationship between contact structures and open book decompositions on 3-manifolds (see [1]). It leads to some topological applications of contact geometry. While reading about the open book decomposition, I came across a result due to Alexander. Its formal statement is
Theorem 1 (Alexander [2]). Every closed oriented 3-manifold admits an open book decomposition.
Before going through its proof, let me briefly revisit the concept of an open book decomposition. From now on, I always assume that $M$ is a closed, oriented 3-manifold.
Definition 1. An open book decomposition (OBD) of $M$ is a pair $(B,\pi)$ satisfying two following conditions.
- $B$ is an oriented link in $M$ called the binding of the open book.
- A fibration $\pi: M \setminus B \rightarrow S^1$ such that each fiber $\pi^{-1}(\theta)$ is the interior of a compact surface $\Sigma_{\theta} \subset M$, which is called a page, and $\partial \Sigma_{\theta}=B$ for all $\theta \in S^1$.
Let's look at one of the most fundamental examples of an OBD on $S^3$. Assume $S^3$ is embedded in $\mathbb C^2$, and $(z_1, z_2)= (r_1 e^{I\theta_1}, r_2 e^{ i\theta_2})$ are coordinates on $\mathbb C^2$. Let $U$ be an unknot in $S^3$ defined by $U:= \{z_1 =0\} = \{r_1=0\} \subset S^3$. The fibration $\pi_U: S^3 \setminus U \rightarrow S^1: (z_1,z_2) \mapsto \frac{z_1}{|z_1|}$ gives an OBD for $S^3$.
There is a nice way to visualize this OBD. Since $S^3$ is the compatification of $\mathbb R^3$, the unknot $U$ can be viewed as a straight line in $\mathbb R^3$. We attach $U$ to half-planes along their boundaries and through the compactification, these half-planes transform into the pages of the OBD.
Returning to Theorem 1, its proof relies heavily on two following facts about 3-manifolds.
Proposition 1 (Alexander [2]). Every closed, oriented manifold is the branched cover of $S^3$ over some link $L \subset S^3$.
Proposition 2 (Alexander [2]). Every link in $S^3$ can be braided about the unknot.
In terms of the second fact, a link $L \subset S^3$ is braided about the unknot means that $L$ can be isotoped so that $L \subset S^1 \times D^2=S^3 \setminus U$ and is transverse to pages $\{p\} \times D^2$ for all $p \in S^1$.
Let's call the cover in Proposition 1 by $P: M \rightarrow S^3$. First, define the binding $B: = P^{-1}(U)$. Since $P$ lifts pages of $S^3$ to pages of $M$, the map $\pi =\pi_U \circ P: M \setminus B \rightarrow S^1$ provides the fibering of $M \setminus B$.
The proof seems to be very elegant and straightforward, yet it still leaves me pondering about the actual role of the link $L$ in the proof. Does it play any role beyond being the branch locus of the covering $P$? And why is the transversality condition of $L$ necessary?
In the remainder of this post, I am not going into details of the proof but instead attempting to interpret the necessity of the transversality condition for the link $L$. I am very grateful for the insightful discussion with Professor Nathan Dunfield on this topic.
Assume that $L$ fails to be transverse to every page in $S^3$. Then there exists a page, say $P_1$ where one component of $L$ is tangent to $P_1$. In this case, there might be pages $P_0$ and $P_2$ just below and above $P_1$, respectively, with different homotopy types, as the number of intersection points with $L$ of $P_0$ compared to $P_2$ increases by 2 (see the figure below). Away from the branch locus, the lifting of fibers, namely $P_0$ and $P_2$, are not homotopy equivalent. Therefore, $\pi$ fails to be a fibration.
Another reason might come from the tangency of $L$ and $P_0$ itself. We can think of what happens when lifting into $M$ as a double cover of $S^3$. Upstairs, the preimage of $P_1$ is no longer a surface as it would be singular at the preimage of the tangency of $L$ and $P_0$.
Reference
[1] Giroux E. Géométrie de contact: de la dimension trois vers les dimensions supérieures. In: Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002): 405–414. Beijing: Higher Ed Press; 2002.
[2] Alexander JW. Note on Riemann spaces. Bull Am Math Soc. 1920;26(8):370-372.