A generalization of Thom’s transversality theorem

Vokřínek, Lukáš. "A generalization of Thom’s transversality theorem." Archivum Mathematicum 044.5 (2008): 523-533. <http://eudml.org/doc/250305>.

@article{Vokřínek2008,
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.},
author = {Vokřínek, Lukáš},
journal = {Archivum Mathematicum},
keywords = {transversality; residual; generic; restriction; fibrewise singularity; transversality; residual; generic; restriction; fibrewise singularity},
language = {eng},
number = {5},
pages = {523-533},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {A generalization of Thom’s transversality theorem},
url = {http://eudml.org/doc/250305},
volume = {044},
year = {2008},
}

TY - JOUR
AU - Vokřínek, Lukáš
TI - A generalization of Thom’s transversality theorem
JO - Archivum Mathematicum
PY - 2008
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 044
IS - 5
SP - 523
EP - 533
AB - 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.
LA - eng
KW - transversality; residual; generic; restriction; fibrewise singularity; transversality; residual; generic; restriction; fibrewise singularity
UR - http://eudml.org/doc/250305
ER -