Definable ${\mathcal{C}}^r$ structures on definable topological groups in d-minimal structures

Abstract

Definable topological groups whose topologies are affine have definable ${\mathcal{C}}^r$ structures in d-minimal expansions of ordered fields, where $r$ is a positive integer. We prove this fact using a new notion called partition degree of a definable set. Basic properties of partition degree are also studied.