We prove a generalization of Thom’s transversality theorem. It gives conditions under which the jet map $f_{*}{|}_Y:Y\subseteq J^r(D,M)\to J^r(D,N)$ is generically (for $f:M\to N$) transverse to a submanifold $Z\subseteq J^r(D,N)$. We apply this to study transversality properties of a restriction of a fixed map $g:M\to P$ to the preimage ${\left(j^{s}f\right)}^{-1}\left(A\right)$ 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 ${g|}_{{\left(j^{s}f\right)}^{-1}\left(A\right)}$ is also generic. We also present an example of $A$ where the...

We shall prove the following Theorem. Let Fs and Fu be two continuous transverse foliations with uniformly smooth leaves, of some manifold. If f is uniformly smooth along the leaves of Fs and Fu, then f is smooth.

We prove that the standard action of the mapping class group $\mathrm{Map}\left(\Sigma \right)$ of a surface $\Sigma$ of sufficiently large genus on the unit tangent bundle $T^1\Sigma$ is not homotopic to any smooth action.