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

TL;DR

This paper investigates the topological properties of definable sublevel sets using o-minimal structures, establishing continuity and deformation retraction results, and explores their implications for topological data analysis transforms.

Contribution

It introduces new theoretical results on the Euler characteristic and homotopy types of definable sublevel sets, linking these to topological data analysis methods.

Findings

Euler characteristic of sublevel sets is right-continuous

Sublevel sets deformation retract for small perturbations

Connections between ECT, smooth ECT, ERT, and smooth ERT established

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 $S^f_{t+\delta}$ deformation retracts to $S^f_t$ for all sufficiently small $\delta > 0$. 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.