Lefschetz fibrations on cotangent bundles and some plumbings

TL;DR

The work provides explicit algorithms to construct Lefschetz fibrations on Weinstein manifolds arising as cotangent bundles and certain plumbings, directly from Weinstein handle decompositions. By leveraging product-structure of subcritical parts and Legendrian data from attaching spheres, it yields regular fibers and vanishing cycles through an injective attaching framework, with inductive constructions and explicit examples. The results extend to plumbing spaces and tree-plumbings, yielding families of diffeomorphic manifolds and new viewpoints on Milnor fibers via Lefschetz data. Collectively, the paper advances a practical method to translate Weinstein structures into Lefschetz fibrations, with potential applications to Fukaya–Seidel categories, symplectic monodromy, and diffeomorphism classifications of complex-analytic singularity fibers.

Abstract

We introduce an idea of constructing Lefschetz fibrations of Weinstein manifolds from Weinstein handle decompositions on them. We prove theorems that formulate the idea for the cases of cotangent bundles and some plumbings. As a corollary, we give diffeomorphic families of plumbing spaces. Those diffeomorphic families contain some plumbing spaces with names. For example, Milnor fibers of $A_{4k+3}$ and $D_{4k+3}$ singularities are diffeomorphic if their complex dimension is odd.

Figures (28)

• Figure 1: The left is a $3$-dimensional $1$-handle $h$, and the right is a division of $h$ into a $3$-handle $h^3$ (red), a $2$-handle $h^2$ (blue), and the other $1$-handle $h^{ori}$ (complement of red and blue). One can observe that the red and blue handles are in a canceling pair.
• Figure 2: a). An example of handle decomposition $D$ of an annulus with an index $0$-handle $h_0$ and an index $1$-handle $h_1$. b). A handle decomposition $\tilde{D}$ induced from $D$. c). The Lefschetz fibration $\pi_0$. We note that the zero sections of two fiber $\pi_0^{-1}(\pm 1)$ are identified with $\partial (h_0^1 \cup h_0^{ori}) = \partial h_0 \cup \partial h_0^2$. d). The projected image of the Legendrian that we would like to achieve by Legendrian isotoping. We note that the colored parts in d) correspond to the same colored part in b). The red parts correspond to $\Lambda_1^{sub}$ and the blue part corresponds to $\Lambda_1^{cri}$.
• Figure 3:
• Figure 4: a) The square, both sides (resp. the top and the bottom) are identified to each other, is the torus. The torus is decomposed into one $0$-handle $h_0$ (center circle), two $1$-handles $h_1, h_2$ whose boundaries are red and blue lines respectively, and one $2$-handle $h_3$ (the rest). b) It describes the induced handle decomposition $\tilde{D}$ of a torus when $D$ is the given decomposition in a). In other words, for $i=1,2$, an $1$-handle $h_i$ is divided into two $1$-handles $h_i^{ori}, h_i^1$ and one $2$-handle $h_i^2$.
• Figure 5: a) describes $M_0$, i.e., union of $h_0^{ori}$ and $h_0^1$. Similarly, in b), c), and d) describe $M_1, M_2$ and $M_3$, respectively. For each $M_i$, the labeled handles are in $M_i \setminus M_{i-1}$

Theorems & Definitions (83)

• Theorem 1.1: Technical statement is Theorem \ref{['thm main']}
• Theorem 1.2: Technical statement is Theorem \ref{['thm plumbing']}
• Theorem 1.3: Technical statement is Theorem \ref{['thm plumbing along tree']}
• Remark 1.4
• Remark 1.5
• Proposition 1.6: Technical statement is Proposition \ref{['prop handle moves']}
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Definition 2.4