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

TL;DR

This paper demonstrates that definable topological groups with affine topologies in d-minimal structures can be endowed with definable $ ext{C}^r$ structures, introducing the novel concept of partition degree to achieve this.

Contribution

The paper introduces the concept of partition degree and proves that such groups admit definable $ ext{C}^r$ structures, advancing the understanding of smooth structures in d-minimal settings.

Findings

Definable topological groups with affine topology have definable $ ext{C}^r$ structures.

Partition degree is a useful tool for analyzing definable sets.

Basic properties of partition degree are established.

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.