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A New Explicit Way of Obtaining Special Generic Maps Into the 3 A NEW EXPLICIT WAY OF OBTAINING SPECIAL GENERIC MAPS INTO THE 3-DIMENSIONAL EUCLIDEAN SPACE NAOKI KITAZAWA Abstract. A special generic map is a smooth map regarded as a natural gen- eralization of Morse functions with just 2 singular points on homotopy spheres. Canonical projections of unit spheres are simplest examples of such maps and manifolds admitting special generic maps into the plane are completely deter- mined by Saeki in 1993 and ones admitting such maps into general Euclidean spaces are determined under appropriate conditions. Moreover, if the difference of dimensions of source and target manifolds are not so large, then the diffeomorphism types of source manifolds are often limited; for example, homotopy spheres except standard spheres do not admit special generic maps into Euclidean spaces whose dimensions arenot so low. As another example, 4-dimensional simply connected manifolds admitting special generic maps are only manifolds represented as the connected sum of the total spaces of 2-dimensional sphere bundles over the 2-dimensional sphere (or the 4-dimensional standard sphere) and there are many manifolds homeomorphic and not diffeomorphic to these manifolds. These explicit facts make special generic maps attractive objects in the theory of Morse functions and higher dimensional versions and application to algebraic and differentiable topology of manifolds, which is an important study in both singuarity theory of maps and algebraic and differential topology of manifolds. In this paper, we demonstrate a way of construction of special generic maps into the 3-dimensional Euclidean space. For this, first we prepare maps onto 2-dimensional polyhedra regarded as simplicial maps naturally called pseudo quotient maps, which are generalizations of the quotient maps to the spaces of all the connected components of inverse images, so-called Reeb spaces of original smooth maps, being fundamental and important tools in the studies. More precisely, we prepare specific pseudo quotient maps, locally construct maps onto 3-dimensional manifolds, glue them and realize the map as a special generic map into the 3-dimensional space. Most of the procedure for the construction is based on technique the author previously noticed and used in simpler situations. arXiv:1806.04581v4 [math.GT] 29 Nov 2018 The success of the construction explicitly shows that a class of maps which seems to cover a larger class of source manifolds may not be not so large and that the diffeomorphism types of source manifolds may be restricted as strongly as in the case of special generic maps. We also explain differential topological facts and problems related to this. 1. Introduction. 1.1. Backgrounds and fundamental tools. Morse functions and higher dimen- sional versions and application to algebraic and differentiable topology of manifolds 2010 Mathematics Subject Classification. Primary 57R45. Secondary 57N15. Key words and phrases. Singularities of differentiable maps; generic maps. Differential topology. 1 2 NAOKI KITAZAWA is an important study in both singularity theory of maps and algebraic and differ- ential topology of manifolds. A fold map is a smooth map such that each singular point is p is of the form m−i(p) 2 m 2 (x1, ··· , xm) 7→ (x1, ··· , xn−1, Pk=n xk − Pk=m−i(p)+1 xk ) for some integers m−n+1 m,n,i(p). i(p) is taken as a non-negative integer not larger than 2 uniquely and we call i(p) the index of p. The set of all the singular points of an index is a smooth submanifold of dimension n−1. Morse functions are regarded as fold maps. A fold map is stable or the C∞ equivalence classes of the maps are invariant under slight perturbations in the C∞ Whitney topology, if and only if the restriction to the set of all the singular points (of a fixed index), which are codimension 1 smooth immersions, are transversal. Note that for example, stable Morse functions or Morse functions such that at distinct singular points, the values are distinct, exist densely. For Morse functions, fold maps and stable maps etc., see [3] for example. For algebraic and differential topological properties of fold maps, see [22] for example. Special generic maps are fold maps indices of whose singular points are 0. They are regarded as simplest generalizations of Morse functions with just 2 singular points on homotopy spheres. Canonical projections of unit spheres are simplest examples of such maps and manifolds admitting special generic maps into the plane are completely determined by Saeki in 1993 [23], ones admitting such maps into general Euclidean spaces are determined under appropriate conditions and more precise facts have been shown. For example, if the difference of dimensions of source and target manifolds are not so large, then the diffeomorphism types of source manifolds are often limited; homotopy spheres except standard spheres do not admit special generic maps into Euclidean spaces whose dimensions are not so low. As another example, 4-dimensional simply connected and closed manifolds admitting special generic maps into R3 are only manifolds represented as the connected sum of the total spaces of smooth S2-bundles over the 2-dimensional sphere (or the 4- dimensional standard sphere) and there are many manifolds homeomorphic and not diffeomorphic to these manifolds; in [27] and [28], there are several 4-dimensional closed manifolds admitting fold maps into R3 and admitting no special generic maps into R3 homeomorphic to ones admitting special generic maps into R3. These facts make special generic maps attractive objects from the viewpoint of differential topology of smooth manifolds. 1.2. Contents of this paper. We define a normal spherical fold map. Definition 1. A stable fold map from a closed manifold of dimension m into Rn satisfying m ≥ n is said to be normal spherical (standard-spherical) if the inverse image of each regular value is a disjoint union of (resp. standard) spheres (or points) and the connected component containing a singular point of the inverse image of a small interval intersecting with the singular value set at once in its interior is either of the following. (1) The (m − n + 1)-dimensional standard closed disc. (2) A manifold PL homeomorphic to an (m − n + 1)-dimensional compact manifold obtained by removing the interior of three disjoint (m − n + 1)-dimensional smoothly embedded closed discs from the (m − n + 1)- dimensional standard sphere. OBTAINING SPECIAL GENERIC MAPS INTO THE 3-DIMENSIONAL SPACE 3 As a fundamental fact, this class includes special generic maps. Such fold maps are studied by Saeki and Suzuoka for example and for example, they can be ob- tained by projections of special generic maps by fundamental discussions of Saeki and Suzuoka [29]. Conversely, whether manifolds admitting such maps admit spe- cial generic maps into one dimensional higher Euclidean spaces is a natural and interesting problem, explicitly stated in the paper. Several answers have been given in [10] and [11] in the case of Morse functions. In this paper, we consider such maps and extended ones and by applying technique based on ideas useful for obtaining the shown theorems for several cases of Morse functions, we obtain special generic maps. As additional remarks, we explain about diffeomorphsim types of source manifolds admitting such maps: for example, we will present the discovered fact that classes of source manifolds are not so large as conjectured and related topics on differential topology of manifolds. This paper is organized as follows. In the next section, we review Reeb spaces and we define pseudo special generic maps and as general PL maps, pseudo quotient maps, which were first introduced in [12]. Pseudo special generic maps have been introduced also in [9] and [11]. They are fundamental objects in the paper. After introducing these maps, we perform construction of special generic maps into R3; this is a main result. Last, we explain remarks on differential topological meanings of the obtained result and related facts and problems. Throughout the paper, all the manifolds and maps between them and bundles over manifolds with fibers being manifolds etc. are smooth and of class C∞ un- less otherwise stated. We also note that maps between polyhedra are PL unless otherwise stated in the paper. Last, we call the set of all singular points of a smooth map the singular set of the map and we call the image of the singular set the singular value set. We call the set of all regular values the regular value set. 2. Reeb spaces, triangulable maps and two classes of maps on manifolds. 2.1. Reeb spaces and triangulable maps. We review Reeb spaces. For precise facts, see also [21] and introduced papers. Definition 2. Let X and Y be topological spaces. For points p1,p2 ∈ X and for a continuous map c : X → Y , we define as p1∼cp2 if and only if p1 and p2 are in the same connected component of c−1(p) for some p ∈ Y . The relation is an equivalence relation and we call the quotient space Wc := X/∼c the Reeb space of c. Example 1. (1) For a Morse function, the Reeb space is a graph. (2) For a special generic map, the Reeb space is regarded as an immersed compact manifold with a non-empty boundary whose dimension is same as that of the target manifold. The Reeb space of a special generic map is known to be contractible if and only if the source manifold is a homotopy sphere. It is also a fundamental and important property that the quotient map from the source manifold onto the Reeb space induces isomorphisms on homology groups whose degrees are not larger than the difference of the dimension of the source manifold and that of the target one.
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