发表机构
Joint Graduate School of Mathematics for Innovation, Kyushu University（九州大学数学创新联合研究生院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为带边流形上的 $\partial$-Morin 映射建立粘合框架，推导欧拉示性数公式、符号差同余及奇异纤维与单纯体积的不等式，并应用于存在性与非奇异延拓问题。
AI 中文摘要
我们研究 $\partial$-Morin 映射，这是一类从带边流形出发的光滑映射，要求映射本身及其在边界上的限制都只有 Morin 奇点，并且映射的奇点集与边界不相交。我们发展了一种粘合构造，为通过闭流形上的相应映射来研究带边流形上的映射提供了统一框架。利用该框架，我们推导了 $\partial$-Morin 映射的欧拉示性数公式，以及从 $4$ 维流形到 $3$ 维流形的 $\partial$-折叠映射中，符号差与奇点集自交数之间的同余关系。我们还建立了从 $3$ 维流形到平面的稳定映射的奇异纤维与源流形的单纯体积之间的不等式。我们进一步将这些结果应用于 $\partial$-折叠映射的存在性问题以及非奇异延拓问题。
英文摘要
We study $\partial$-Morin maps, a class of smooth maps from manifolds with boundary such that both the maps themselves and their restrictions to the boundary have only Morin singular points, and the singular point sets of the maps are disjoint from the boundary. We develop a gluing construction that provides a unified framework for studying maps from manifolds with boundary through corresponding maps from closed manifolds. Using this framework, we derive Euler characteristic formulas for $\partial$-Morin maps and a congruence relating the signature to the self-intersection number of the singular point set for $\partial$-fold maps from $4$-manifolds to $3$-manifolds. We also establish an inequality relating singular fibers of stable maps from a $3$-manifold to the plane to the simplicial volume of the source manifold. We further apply these results to the existence problem for $\partial$-fold maps and to the non-singular extension problem.
Comments36pages, 7 figures, 1 table
