Lefschetz fibrations on cotangent bundles and some plumbings

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.