Simplifying indefinite fibrations on 4-manifolds

Abstract

The main goal of this article is to connect some recent perspectives in the study of 4 4 -manifolds from the vantage point of singularity theory. We present explicit algorithms for simplifying the topology of various maps on 4 4 -manifolds, which include broken Lefschetz fibrations and indefinite Morse 2 2 -functions. The algorithms consist of sequences of moves, which modify indefinite fibrations in smooth 1 1 -parameter families. These algorithms allow us to give purely topological and constructive proofs of the existence of simplified broken Lefschetz fibrations and Morse 2 2 -functions on general 4 4 -manifolds, and a theorem of Auroux–Donaldson–Katzarkov on the existence of certain broken Lefschetz pencils on near-symplectic 4 4 -manifolds. We moreover establish a correspondence between broken Lefschetz fibrations and Gay–Kirby trisections of 4 4 -manifolds, and show the existence and stable uniqueness of simplified trisections on all 4 4 -manifolds. Building on this correspondence, we also provide several new constructions of trisections, including infinite families of genus- 3 3 trisections with homotopy inequivalent total spaces, and exotic same genera trisections of 4 4 -manifolds in the homeomorphism classes of complex rational surfaces.

Uniqueness of 4-manifolds described as sequences of 3-d handlebodies

“…Since the path is constant near the boundary circles it can be viewed as a generic path of functions on the original surface Σ, giving the desired homotopy f t for t ∈ [ 2 6 , 5 6 ] thus completing the homotopy f t for t ∈ [0, 1]. Repeating the process we can find a generic path f t for t ∈ [1,2]. Composing the two homotopies gives the desired result.…”

“…We next seek to understand how vanishing cycles change as we modify a Morse 2-function. Following [2], we say a local modification of critical images which takes a critical image C 0 and produces a critical image C 1 is always realizable if there is a smooth 1-parameter family of smooth maps F t : X such that the critical image of F 0 is C 0 and the critical image of F 1 is C 1 . We will need three particular always realizable moves in this paper.…”

Multisections of 4-manifolds

“…In this section we discuss connections to stable maps of 4-manifolds to the disk. The reader is referred to [BS17] for an overview of the topic well suited to our setting and to [GG73] for the theoretical underpinnings of the subject. The smooth Figure 22.…”

Elimination of Definite Fold. Ii

“…Note that such an algorithmic construction is an important ingredient in [3,4] for constructing simplified broken Lefschetz fibrations and simplified trisections on fourdimensional manifolds.…”

“…This made an explicit construction very hard to realize. On the contrary, our new proof presented in this paper modifies the original map only by homotopy: furthermore, it is not only constructive, but also algorithmic, which is an important ingredient for the simplification process for maps on 4-manifolds proposed in [3] (see also [4]).…”