发表机构
Osaka Central Advanced Mathematical Institute（大阪中央高等数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究非固有光滑函数的Reeb空间，推广Saeki的定理，指出闭流形上光滑函数的Reeb空间为图当且仅当其临界值集有限，属相关拓扑与组合研究的进展。
AI 中文摘要
本文讨论非固有光滑函数的Reeb空间。（非）固有函数是拓扑空间之间的映射，其紧集的原像为紧集（对应非紧集的情况则不要求原像为紧集）。它们的Reeb空间是其水平集的连通分支构成的空间，并带有自然的商拓扑。在固有函数的情形，自20世纪50年代以来已发展出明确的理论，这与Morse（-Bott）函数理论密切相关，此时自然得到图结构。近年来，相关的一般拓扑与组合研究正积极推进，主要由Gelbukh和Saeki开展，在 tame 情形下Reeb空间至多为1维。本文推广了Saeki在2020年代提出的定理：闭流形上光滑函数的Reeb空间自然为图当且仅当其临界值集有限。作者是非固有情形研究的先驱，此前已对相关实例进行了探究。
英文摘要
We discuss Reeb spaces of non-proper smooth functions. (Non-)proper functions are maps between topological spaces the preimages of compact sets by which are compact (resp. may not be compact). Their Reeb spaces are the spaces of connected components of their level sets and with the natural quotient topologies. In proper cases, explicit theory are developing since the 1950s. This is closely related to Morse(-Bott) function theory and in such cases we have graphs naturally. Recently, related general topological studies with combinatorial ones are actively developing, mainly due to Gelbukh and Saeki, and they are at most $1$-dimensional in tame cases. We extend a theorem by Saeki in the 2020s: the Reeb space of a smooth function on a closed manifold is naturally a graph if and only if its critical value set is finite. The author has been a pioneer of the non-proper case and previously investigated examples.
Comments8 pages. The author has found small errors and gaps, where it is not so essential in Main Theorem. The author has corrected and revised them
