Transversality of smooth definable maps in O-minimal structures

TL;DR

This paper establishes a definable smooth version of Thom's transversality theorem within o-minimal structures, demonstrating the genericity and stability of transversality for definable smooth maps.

Contribution

It introduces a definable smooth transversality theorem, showing the set of non-transverse maps is nowhere dense, and extends Trotman's stability theorem to the definable setting.

Findings

Non-transverse maps form a nowhere dense set

Transversality is generically stable in the definable smooth topology

Whitney (a)-regularity characterizes transversality stability

Abstract

We present a definable smooth version of the Thom transversality theorem. We show further that the set of non-transverse definable smooth maps is nowhere dense in the definable smooth topology. Finally, we prove a definable version of a theorem of Trotman which says that the Whitney $(a)$-regularity of a stratification is necessary and sufficient for the stability of transversality.