A generalization of Thom’s transversality theorem. (English) Zbl 1212.57010

Summary: We prove a generalization of Thom’s transversality theorem. It gives conditions under which the jet map \(f_{\ast }| _Y\: Y\subseteq J^r(D,M)\to J^r(D,N)\) 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 \((j^sf)^{-1}(A)\) 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.


57R35 Differentiable mappings in differential topology
57R45 Singularities of differentiable mappings in differential topology
58A20 Jets in global analysis
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