佐伯 修

This chapter describes how differentiable maps of manifolds into Euclidean spaces with singularities are related to the topological or differentiable structures of manifolds. Singularities of differentiable maps are formulated locally in principle: however, maps with certain singularities as a whole or the singularities in total carry global information. The study of differentiable maps with singularities from such kind of a viewpoint is called the global singularity theory of differentiable maps. In this chapter, we first focus on differentiable maps with only definite fold singularities, called special generic maps, and see how such maps affect the differentiable structures of the source manifolds. Then, we introduce the notion of cobordisms for maps with prescribed singularities, which will be used to extract certain invariants of singular maps and the source manifolds. We will see that singular fibers play important roles in studying such cobordisms. Finally, we give a brief exposition of a result due to Gromov, which relates the simplicial volume of a manifold with the number of certain singular fibers.

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

The Reeb space of a continuous function is the space of connected components of the level sets. In this paper we first prove that the Reeb space of a smooth function on a closed manifold with finitely many critical values has the structure of a finite graph without loops. We also show that an arbitrary finite graph without loops can be realized as the Reeb space of a certain smooth function on a closed manifold with finitely many critical values, where the corresponding level sets can also be preassigned. Finally, we show that a continuous map of a smooth closed connected manifold to a finite connected graph without loops that induces an epimorphism between the fundamental groups is identified with the natural quotient map to the Reeb space of a certain smooth function with finitely many critical values, up to homotopy.

特異点論の観点から、特異シンプレクティック構造に付随した特異Lefschetz構造の存在や、単純化されたtrisectionの存在を、具体的かつ構成的に証明することに成功し た。