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transversality; residual; generic; restriction; fibrewise singularity
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.
[1] Golubitsky, M., Guillemin, V.: Stable mappings and their singularities. Grad. Texts in Math., Vol. 14, Springer-Verlag, New York-Heidelberg, 1973. DOI 10.1007/978-1-4615-7904-5 | MR 0341518 | Zbl 0294.58004
[2] Hirsch, M. W.: Differential topology. Grad. Texts in Math., No. 33, Springer-Verlag, New York-Heidelberg, 1976. MR 0448362 | Zbl 0356.57001
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