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

Vokřínek, Lukáš

Arch. Math., Brno 44, No. 5, 523-533 (2008).

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.

MSC:

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

Keywords:

transversality; residual; generic; restriction; fibrewise singularity