On subanalytic subsets of real analytic orbifolds

TL;DR

This paper extends the theory of subanalytic sets and maps to real analytic orbifolds, establishing foundational properties and analogs of classical results from manifold theory.

Contribution

It introduces definitions and proves key properties of subanalytic subsets and maps in the orbifold setting, including existence and behavior under images and inverse images.

Findings

Subanalytic subsets can be realized via proper real analytic maps from orbifolds.

Inverse images of subanalytic sets under subanalytic maps are subanalytic.

Proper subanalytic maps preserve the subanalyticity of images.

Abstract

The purpose of this paper is to define semi- and subanalytic subsets and maps in the context of real analytic orbifolds and to study their basic properties. We prove results analogous to some well-known results in the manifold case. For example, we prove that if $A$ is a subanalytic subset of a real analytic quotient orbifold $X$, then there is a real analytic orbifold $Y$ of the same dimension as $A$ and a proper real analytic map $f\colon Y\to X$ with $f(Y)=A$. We also study images and inverse images of subanalytic sets and show that if $X$ and $Y$ are real analytic orbifolds and if $f\colon X\to Y$ is a subanalytic map, then the inverse image $f^{-1}(B)$ of any subanalytic subset $B$ of $Y$ is subanalytic. If, in addition, $f$ is proper, then also the image $f(A)$ of any subanalytic subset $A$ of $X$ is proper.