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.
@article{127118,
author = {Luk\'a\v s Vok\v r\'\i nek},
title = {A generalization of Thom's transversality theorem},
journal = {Archivum Mathematicum},
volume = {044},
year = {2008},
pages = {523-533},
zbl = {1212.57010},
mrnumber = {2501582},
language = {en},
url = {http://dml.mathdoc.fr/item/127118}
}
Vokřínek, Lukáš. A generalization of Thom’s transversality theorem. Archivum Mathematicum, Tome 044 (2008) pp. 523-533. http://gdmltest.u-ga.fr/item/127118/
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