定义完整的 d 最小结构中的莫尔斯理论

arXiv - MATH - Logic Pub Date : 2024-08-26 DOI:arxiv-2408.14675

Masato Fujita, Tomohiro Kawakami

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引用次数: 0

摘要

考虑一个有序域 $F$ 的可定义完整 d 最小展开 $(F, <, +, \cdot, 0, 1,\dots,)$ 。让 $X$ 是一个可定义的紧凑可定义的正常可定义的 $C^r$ 流形，且 $2 \le r <\infty$.我们证明，就可定义的 $C^2$ 拓扑而言，在 $X$ 上可定义的 $C^r$ 函数集合中，可定义的莫尔斯函数集合是开放且密集的。

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Morse theory in definably complete d-minimal structures

Consider a definable complete d-minimal expansion $(F, <, +, \cdot, 0, 1, \dots,)$ of an oredered field $F$. Let $X$ be a definably compact definably normal definable $C^r$ manifold and $2 \le r <\infty$. We prove that the set of definable Morse functions is open and dense in the set of definable $C^r$ functions on $X$ with respect to the definable $C^2$ topology.

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arXiv - MATH - Logic

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