三维欧几里得空间特殊一般映射的一种新的显式方法
A new explicit way of obtaining special generic maps into the 3-dimensional Euclidean space
摘要：
一种特殊的一般映射是光滑映射，它被看作是同伦球面上仅有2个奇点的Morse函数的自然推广。单位球面的正则投影是这种映射的最简单的例子，Saeki在1993年完全确定了允许特殊一般映射进入平面的流形，并在适当的条件下确定了允许这种映射进入一般欧几里得空间的流形。此外，如果源流形和目标流形的维数相差不大，那么源流形的同构类型往往是有限的。这些明确的事实使特殊的一般映射在Morse函数理论和高维版本中成为引人注目的对象，并应用于流形的代数和可微拓扑，这在映射的奇异性理论和流形的代数和微分拓扑中都是一个重要的研究。本文给出了三维欧几里得空间中特殊一般映射的一种构造方法。为此，我们首先准备到二维多面体上的映射，这些映射被视为单纯映射，自然称为伪商映射，它们是商映射到逆像的所有连通分量的空间的推广，即所谓的原始光滑映射的Reeb空间，是研究中的基本和重要工具。构造的成功明确表明，一类映射，似乎覆盖了一个更大的类的源流形可能不是那么大，源流形的同构类型可能受到严格的限制，作为特殊的一般映射。我们还解释了微分拓扑事实和与此相关的问题。
英文摘要：
A special generic map is a smooth map regarded as a natural generalization 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 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. 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. 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 special generic maps. We also explain differential topological facts and problems related to this.
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