由流形通过Reeb空间导出的光滑映射正则值原像的附加结构及其应用
Investigating additional structures on preimages of regular values of smooth maps induced from the manifolds via the Reeb spaces and applications
摘要：
一般映射的Reeb空间是映射的原像的所有连通分量的空间。Reeb空间是Morse函数和高维变式理论及其在几何中的应用的基础和有用的工具。这些空间是多面体兼容的自然单纯结构的流形在相当多的情况下。例如，对于Morse函数、折叠映射和适当稳定映射，它们在这些研究中是有用的和基本的工具，它们是多面体，其尺寸等于目标的流形的尺寸。在2010年代，Hiratuka和Saeki发现Reeb空间的顶维同调群对于光滑映射（例如具有一些包含非（定向）零协边且没有奇点的分量的原像的真稳定映射）不会为零。最近，作者发现了这一事实的一些扩展版本，制定和使用光滑和闭流形的等价类的类配边群。本文研究了不含映射奇点的流形和原像除了可微结构和定向外，还可能具有额外的代数拓扑或微分拓扑结构的情形。我们证明了这种情况下的定理。我们还给出了一些显式的应用，例如，原像具有自旋或自旋^c结构，从域的流形和框架流形的结构诱导。作为重要的应用，我们给出了自旋配边和自旋^c配边的新的显式构造。
英文摘要：
The Reeb space of a generic map is the space of all connected components of preimages of the map. Reeb spaces are fundamental and useful tools in the theory of Morse functions and higher dimensional variants and their applications to geometry. These spaces are polyhedra compatible with natural simplicial structures of the manifolds in considerable cases. For example, for Morse functions, fold maps, and proper stable maps, which are useful and fundamental tools in these studies, they are polyhedra whose dimensions are equal to those of the manifolds of the targets. In the 2010s, Hiratuka and Saeki found that the top-dimensional homology group of the Reeb space does not vanish for smooth maps such as proper stable maps having some preimages containing a component which is not (oriented) null-cobordant and which has no singular points. Recently the author found some extended versions of this fact formulating and using cobordism-like groups of equivalence classes of smooth and closed manifolds. The present paper concerns cases where manifolds and preimages with no singular points of maps may have additional algebraic topological or differential topological structures other than differentiable structures and orientations. We show a theorem for this case. We also give some explicit applications of the result where the preimages have spin or spin^c structures induced from the manifolds of the domains and structures of framed manifolds for example. As important applications, we present new explicit construction of spin cobordisms and spin^c cobordisms for example.
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