Both Reeb spaces and higher Morse functions induce natural stratifications. In the former, we show that the data of the Jacobi set of a function $f:X \to \mathbb{R}^k$ induces stratifications on $X,\mathbb{R}^k$, and the associated Reeb space, and give conditions under which maps between these three spaces are stratified maps. We then extend this type of construction to the codomain of higher Morse functions, using the singular locus to induce a stratification of which sub-posets are equivalent to multi-parameter filtrations.

中文翻译：

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Reeb 空间和高莫尔斯函数的自然分层

Reeb 空间和更高的 Morse 函数都会引起自然分层。在前者中，我们表明函数 $f:X \to \mathbb{R}^k$ 的 Jacobi 集的数据在 $X、\mathbb{R}^k$ 和相关的 Reeb 空间上引起分层，并给出这三个空间之间的映射是分层映射的条件。然后，我们将这种类型的构造扩展到更高莫尔斯函数的余域，使用奇异轨迹来诱导分层，其中子姿势等效于多参数过滤。