Lefschetz Fibrations (4/25/26)
Given a 3-dimensional contact manifold \((M, \xi)\), it is of interest to characterize the symplectic 4-manifolds \((W,\omega)\) for which \(\partial W = M\). Per Giroux, every contact structure on a closed, oriented 3-manifold is supported by an open book. It is then most natural to then consider a fibration that will induce an open book decomposition on \(\partial W\). [An open book decomposition of \(M\) is a fibration \(\pi: M\backslash B \rightarrow S^1\) for \(B \subset M\) an oriented link called the binding with fibers the interior of a compact surface \(\Sigma_t \subset M\) called pages, such that \(\partial \Sigma_t = B\).]
For \(W\) a compact, oriented 4-manifold with boundary and corners, let \(\partial W = \partial_h W \cup \partial_v W\), decomposing the boundary into horizontal and vertical parts. By horizontal, we generally mean the boundary of the fibers over the base and by vertical we mean the fibers over the boundary of the base. For instance, the trivial fibration \(\pi: T^2 \backslash \{*\} \times D^2 \rightarrow D^2\) admits the decomposition \(\partial(T^2 \backslash \{*\} \times D^2) = \partial (T^2 \backslash \{*\}) \times D^2 \hspace{1mm} \cup \hspace{1mm} T^2 \backslash \{*\} \times \partial D^2\) with corner \(* \times \partial D^2\). Then, a bordered Lefschetz fibration of W over \(\mathbb{D} \subset \mathbb{C}\) is a smooth map \(\pi: W \rightarrow \mathbb{D}\) such that:
- \(\pi\) has finitely many critical points in \(\text{int}(W)\) and critical values in \(\text{int}(\mathbb{D})\).
- Near each critical point \(p\), there are complex coordinates \((z_1,z_2)\) compatible with the orientation of \(W\) such that \(\pi(z_1,z_2) = z_1^2 + z_2^2\) on a neighborhood \(U \supset p\).
- \(\partial \pi^{-1}(z) \neq \emptyset\).
- \(\partial_v W = \pi^{-1}(\partial \mathbb{D})\) with \(\pi |_{\partial_v W} : \partial_v W \rightarrow \partial \mathbb{D}\) a smooth fibration.
- \(\partial_h W = \cup_{z \in \mathbb{D}} \partial (\pi^{-1}(z))\) with \(\pi |_{\partial_h W}: \partial_h W \rightarrow \mathbb{D}\) a smooth fibration.
We call the fibers corresponding to regular values regular fibers. Otherwise, they are singular fibers. Additionally, we may define a vanishing circle on the structure of a Lefschetz fibration. Consider some smooth path \(\gamma_z: I \rightarrow \mathbb{D}\) for each critical value \(z\) such that \(\gamma_z(0)=z_0 \notin \mathbb{D}^{\text{crit}}, \gamma_z(1) = z\). Then, some (usually homologically nontrivial) smooth embedded circle \(C_p \subset F = \pi^{-1}(z_0)\) can be chosen such that it vanishes under parallel transport along \(\gamma_{\pi(p)}\). Within \(W\), the cone of a vanishing circle over the vanishing point is called a Lefschetz thimble.
Example 1. Consider the 4-ball \(B^4 \subset \mathbb{C}^2\) with \(\pi: B^4 \rightarrow \mathbb{D}\) such that \(\pi(z_1,z_2) = z_1z_2\). As \(\frac{\partial \pi}{\partial z_1} = \frac{\partial \pi}{\partial z_2} = 0\) specifically at \((0,0)\), we have a singular fiber \(\pi^{-1}(0) = \{(0,z_2)||z_2|^2 \leq 1\} \cup \{(z_1,0)||z_1|^2 \leq 1\} \) that is a cone with vertex \((0,0)\) whose boundary is a Hopf link in \(S^3\). To see why we should get a Hopf link between these boundary components, imagine moving from the boundary of one component to the center of its disk. The modulus of that nonzero \(z_1\) should be shrinking to 0, allowing the other complex coordinate \(z_2\) to have modulus approaching 1, which approaches the boundary of the other component. The boundaries thus interlock. The regular fibers are described by \(\pi^{-1}(re^{i\theta})=\{(r_1e^{i\theta_1}, r_2e^{i\theta_2})|r=r_1r_2, \theta = \theta_1 + \theta_2\}\), which appear as annuli (depicted below as cylinders) with boundary a Hopf link.
Figure 2. A bordered Lefschetz fibration.
The definition of a bordered Lefschetz fibration also admits a monodromy action \(\varphi\) on the regular fibers \(F\) that is trivial near \(\partial F\). Consider some \(\gamma: S^1 \rightarrow \mathbb{D}\) that avoids critical values with \(F = \pi^{-1}(\gamma(0))\). Then, \(\forall p \in F\), we can lift \(\gamma\) to \(\tilde{\gamma}_p: I \rightarrow W\) such that \(\varphi_{\gamma}(p) = \tilde{\gamma}_p(1) \in F\). Since \(\mathbb{D}^{\text{crit}} \subset \text{int}(\mathbb{D})\), the set of critical values, \(\varphi\) exists on \(\pi^{-1}(\partial \mathbb{D})\).
We are now prepared to give an open book decomposition of \(M\) via the Lefschetz fibration on \(W\). By \(\mathbb{D}\) contractible, fibers over it must be trivial. As each fiber \(\pi^{-1}(z)\) is a smooth surface with boundary some disjoint union of \(k\) \(S^1\) bundles over \(\mathbb{D}\), \(\partial_h W = \cup_{z \in \mathbb{D}} \partial(\pi^{-1}(z)) = \bigsqcup\limits_{i=1}^{k}(S^1 \times \mathbb{D})\). Now, let \(B = \partial(\pi^{-1}(0)) \subset \partial_h W\) be a \(k\)-component link such that \(\partial_h W\) is a tubular neighborhood of \(B\) in \(\partial W\). Rounding the corners of \(\partial W\) to get the smooth manifold \(M\), we have an open book decomposition of \(M\) per the fibration \(\pi: M \backslash B \rightarrow S^1\) determined by \(\pi|_{\partial_v W} : \partial_v W \rightarrow \partial \mathbb{D}\). Any page of this open book is given by a regular fiber in the Lefschetz fibration, and monodromy of the open book is given by monodromy along \(\partial \mathbb{D}\).
Figure 3. Corner smoothing and the open book decomposition for Example 1.
In fact, we have this wonderful result that relates the monodromy of fibers to the open book decomposition of \(M\).
Theorem 1. If \(\pi: W \rightarrow \mathbb{D}\) is a bordered Lefschetz fibration, then the monodromy \(\varphi: F \rightarrow F\) of the induced open book at the boundary is a composition of positive Dehn twists along the vanishing circles \(C_p \subset F\).
In Example 1, as there is a single vanishing point (labelled), the monodromy of our open book at the boundary is given by a positive Dehn twist along this circle. We call a bordered Lefschetz fibration allowable if none of the vanishing circles are homologically trivial.
Example 2. Consider the bordered Lefschetz fibration with \(F = T^2 \backslash D^2\) such that there are two critical points, each with local monodromy described by a single Dehn twist in the lift. As \(\pi_1(T^2 \backslash D^2) \simeq \mathbb{Z}^2\), the monodromy is described by a 2×2 matrix that maps vanishing circles to each other linearly per the Lefschetz-Picard formula: \(x \mapsto x + (x \cdot \gamma)\gamma\) for \((x \cdot \gamma)\) the algebraic intersection number. For \(a,b\) generating \(\pi_1(T^2 \backslash D^2)\), for a positive Dehn twist about a, \(a \mapsto a + (a \cdot a) a = a\) and \(b \mapsto b + (b \cdot a)a = b + (\pm 1)a\). Using positive convention, we choose \(+1\) to get the monodromy \(\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}\). For a positive Dehn twist about b, \(b \mapsto b\) and \(a \mapsto a + (a \cdot b)b = a-b\) so the monodromy is given by\(\begin{pmatrix} 1 & 0 \\ -1 & 1 \end{pmatrix}\). As these matrices don’t commute, it is clear that Dehn twists generally don’t commute.
Figure 4. Example 2. The lifts of the paths around the critical points give Dehn twists in different directions, and the monodromy along \(\partial \mathbb{D}\) is given by the composition of these smaller monodromies.
So, Lefschetz fibrations give a nice way to look at contact structures as the boundary of a higher dimensional manifold with some interesting open book decomposition properties. In general, if \(F\) is of dimension \(2n-2\), it is a Liouville domain, a type of compact symplectic manifold with boundary that is nice to work with.
The Nifty Nil (4/1/26)
This Zariski topology picture shows that the nilradical also looks like the intersection of primes of some dense set.
Solution
First, we show that every dense set \(S \subseteq\) Spec(R) must satisfy \(V(\cap_{p \in S} p) = \) Spec(R). Note that \(S\) dense means that \(\overline{S} =\) Spec(R) and \(\overline{S} = \cap_{\text{closed} I \supseteq S} I\). Letting \(V(I) = \{p \in \text{Spec(R)} | I \subseteq p\}\), \(\overline{S} = \cap_{V(J) \supseteq S} V(J)\), as all closed sets take such a form for \(J\) an ideal in the Zariski topology. Then, \(\forall p \in S \subseteq V(J)\), \(J \subseteq p\), so \(J \subseteq \cap_{p \in S} p\) and \(V(\cap_{p \in S} p) \subseteq V(J)\) by the order reversing property of \(V(\cdot)\). As this holds for all ideals \(J\), \(V(\cap_{p \in S} p)\) is closed and must therefore be the closure \(\overline{S}\), so \(S\) dense gives (*) \(V(\cap_{p \in S} p) = \) Spec(R).
Now, consider that by \(R\) commutative, \(\cap_{p \in S} p \supseteq \cap_{p \in \text{Spec(R)}} p = \sqrt{0} = (0)\) as \(R\) a domain gives \(b \in \sqrt{0} \Rightarrow \exists n : b^n = 0 \Rightarrow b = 0 \hspace{2mm} \text{or} \hspace{2mm} b^{n-1} = 0 \Rightarrow \ldots \Rightarrow b = 0\). By (*), we also have \(\cap_{p \in S} p \subset \tilde{p}\) for all \(\tilde{p} \in\) Spec(R) so that \(\cap_{p \in S} p \subseteq \cap_{p \in \text{Spec(R)}} p\). Thus, \(\cap_{p \in S} p = (0)\).
The Zariski topology ends up being a very nice tool by which one can view rings from a point-set perspective. Not only does the notion of closure carry over, but so does connectedness. In fact, if Spec(\(R\)) is disconnected, then \(R\) is necessarily a product of rings, which is quite surprising.