Euler Characteristics and Homotopy Types of Definable Sublevel Sets

09/06/2023

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by   Mattie Ji, et al.

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Given a continuous definable function f: S →ℝ on a definable set S, we study sublevel sets of the form S^f_t = {x ∈ S: f(x) ≤ t} for all t ∈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 relationship between the concepts of Euler characteristic transform and smooth Euler characteristic transform in topological data analysis.