  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.
