# Global smoothing of a subanalytic set

## TL;DR

This paper addresses longstanding questions in real-analytic geometry about smoothing subanalytic sets and transforming proper mappings to have equidimensional fibers, providing new insights and solutions.

## Contribution

It offers simple solutions to two major open problems, showing that while the transformation to equidimensional fibers is not always possible, smoothing is achievable.

## Key findings

- Global smoothing of subanalytic sets is possible.
- Transforming proper mappings to equidimensional fibers generally impossible.
- Positive smoothing results despite negative transformation findings.

## Abstract

We give rather simple answers to two long-standing questions in real-analytic geometry, on global smoothing of a subanalytic set, and on transformation of a proper real-analytic mapping to a mapping with equidimensional fibres by global blowings-up of the target. These questions are related: a positive answer to the second can be used to reduce the first to the simpler semianalytic case. We show that the second question has a negative answer, in general, and that the first problem nevertheless has a positive solution.