A generalization of Thom's transversality theorem

TL;DR

This paper extends Thom's transversality theorem to broader conditions, providing new insights into the generic transversality of jet maps and their restrictions, with applications to submanifold intersections.

Contribution

It generalizes Thom's transversality theorem to include more cases and studies the transversality of restricted maps in jet spaces.

Findings

The jet map is generically transverse under new conditions.

Restrictions of maps to preimages of submanifolds are generically transverse.

An example shows the theorem's limitations.

Abstract

We prove a generalization of Thom's transversality theorem. It gives conditions under which the jet map $f_*|_Y:Y\subseteq J^r(D,M)\ra J^r(D,N)$ is generically (for $f:M\ra N$) transverse to a submanifold $Z\subseteq J^r(D,N)$. We apply this to study transversality properties of a restriction of a fixed map $g:M\ra P$ to the preimage $(j^sf)^{-1}(A)$ of a submanifold $A\subseteq J^s(M,N)$ in terms of transversality properties of the original map $f$. Our main result is that for a reasonable class of submanifolds $A$ and a generic map $f$ the restriction $g|_{(j^sf)^{-1}(A)}$ is also generic. We also present an example of $A$ where the theorem fails.