# Ren{\'e} Thom and an anticipated h-principle

## TL;DR

This paper explores René Thom's contributions to the development of the h-principle, highlighting his early homological version and its relation to Smale's immersion theory and Thurston's jiggling technique.

## Contribution

It provides a historical and mathematical analysis of Thom's homological h-principle and its connection to key ideas in differential topology and foliation theory.

## Key findings

- Thom's homological h-principle was largely ignored by Paris topologists.
- A combinatorial subdivision of the simplex is central to Thom's theorem.
- Thom's ideas connect with Thurston's jiggling technique in foliation theory.

## Abstract

The first part of this article intends to present the role played by Thom in diffusing Smale's ideas about immersion theory, at a time (1957) where some famous mathematicians were doubtful about them: it is clearly impossible to make the sphere inside out! Around a decade later, M. Gromov transformed Smale's idea in what is now known as the h-principle. Here, the h stands for homotopy.Shortly after the astonishing discovery by Smale, Thom gave a conference in Lille (1959) announcing a theorem which would deserve to be named a homological h-principle. The aim of our second part is to comment about this theorem which was completely ignored by the topologists in Paris, but not in Leningrad. We explain Thom's statement and answer the question whether it is true. The first idea is combinatorial. A beautiful subdivision of the standard simplex emerges from Thom's article. We connect it with the jiggling technique introduced by W. Thurston in his seminal work on foliations.