Euler Characteristics and Homotopy Types of Definable Sublevel Sets, with Applications to Topological Data Analysis

Mattie Ji; Kun Meng

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arXiv · 2309.03142

Euler Characteristics and Homotopy Types of Definable Sublevel Sets, with Applications to Topological Data Analysis

Mattie Ji·Kun Meng

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Abstract

Given a definable function $f: S \to \mathbb{R}$ on a definable set $S$, we study sublevel sets of the form $S^f_t \coloneqq \{x \in S: f(x) \leq t\}$ for all $t \in \mathbb{R}$. Using o-minimal structures, we prove that the Euler characteristic of $S^f_t$ is right-continuous with respect to $t$. Furthermore, when $S$ is compact, we show that $S^f_{t+δ}$ deformation retracts to $S^f_t$ for all sufficiently small $δ> 0$. Applying these results, we also characterize the connections between the following concepts in topological data analysis: the Euler characteristic transform (ECT), smooth ECT, Euler-Radon transform (ERT), and smooth ERT.

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BibTeX RIS

Mattie Ji, Kun Meng. 2026-03-25. Euler Characteristics and Homotopy Types of Definable Sublevel Sets, with Applications to Topological Data Analysis. https://arxiv.org/abs/2309.03142

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