A generalization of Thom’s transversality theorem

Lukáš Vokřínek

Address: Department of Mathematics and Statistics, Masaryk University Kotlářská 2, 611 37 Brno, Czech Republic

E-mail: koren@math.muni.cz

Abstract: We prove a generalization of Thom’s transversality theorem. It gives conditions under which the jet map $f_*|_Y\colon Y\subseteq J^r(D,M)\rightarrow J^r(D,N)$ is generically (for $f\colon M\rightarrow N$) transverse to a submanifold $Z\subseteq J^r(D,N)$. We apply this to study transversality properties of a restriction of a fixed map $g\colon M\rightarrow P$ 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.

AMSclassification: primary 57R35; secondary 57R45.

Keywords: transversality, residual, generic, restriction, fibrewise singularity.