佐伯 修

A smooth map between manifolds is said to be image simple if its restriction to its singular point set is a topological embedding. We study the parity of the number of connected components of the singular point set for image simple fold maps from a closed manifold of dimension greater than or equal to 2 to a surface. It is known that this parity is a homotopy invariant when the source manifold is of even dimension and the target surface is orientable. In this paper we show that for an arbitrary image simple fold map from a closed odd-dimensional manifold of dimension greater than or equal to 3 to a (possibly non-orientable) surface, this parity is not a homotopy invariant. We give two constructive proofs: one uses techniques from open book decompositions and round fold maps, and the other uses allowable moves. The constructed homotopies add one new component to the singular point set. We also give a new proof of the homotopy invariance of the parity of the number of connected components of the singular point set for image simple fold maps from a closed even-dimensional manifold to an orientable surface together with an example showing that this does not hold for maps into non-orientable surfaces in general.

One of the most effective and familiar methods to study the topological structure of a given differentiable manifold is to use Morse functions defined on it. Such functions can be regarded as generic differentiable maps into the real line. Then, what happens if we consider generic maps into general dimensional Euclidean spaces or manifolds? This might have been a motivation of Whitney or Thom around the middle of the 20th century for studying singularities of differentiable maps between manifolds. In this survey article, following such an idea, we overview some studies of structures of manifolds by using generic differentiable maps, and some global studies of generic differentiable maps with singularities themselves, including recent developments.

Let F : (R^M, 0) → (R^N, 0) and G : (R^N, 0) → (R^K , 0), M ≥ N ≥ K ≥ 2, be non-constant real analytic map germs with isolated critical values. In this paper we study the topology of the Milnor tube fibrations of the map germs F, G and their composition H = G ◦ F, under the tame condition. More precisely, we show that the Milnor fiber F_H of H is homotopy equivalent to the product F_F × F_G of the Milnor fibers of F and G, and that the boundary ∂F_H of F_H is homotopy equivalent to ∂(F_F × F_G) = (∂F_F × F_G) ∪ (F_F × ∂F_G). Furthermore, if each component of ∂(F_F × F_G) is simply connected and M − K ≥ 6, then the homotopy equivalences can be replaced by diffeomorphisms in the above statements.

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.

We consider links of complex isolated hypersurface singularities in C^{n+1} and study differentiable maps defined by restricting holomorphic functions to the links. We give an explicit example in which such a restriction gives a fold map into the plane C = R^2 whose singular value set consists of concentric circles.

多様体上の可微分関数のReeb空間が、有限グラフの構造を持つための、関数に関する必要十分条件を、レベル集合の連結成分がその近傍をどのように分割するかという観点から与えることに初めて成功した。

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。