A Generalization of Thom’s Transversality Theorem

We prove a generalization of Thom’s transversality theorem. It gives conditions under which the jet map f∗|Y : Y ⊆ Jr(D,M)→ Jr(D,N) is generically (for f : M → N) transverse to a submanifold Z ⊆ Jr(D,N). We apply this to study transversality properties of a restriction of a fixed map g : M → P to the preimage (jsf)−1(A) of a submanifold A ⊆ Js(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|(jsf)−1(A) is also generic. We also present an example of A where the theorem fails. 0. Introduction We start by reminding that for smooth manifolds M and N the set C∞(M,N) of smooth maps is endowed with two topologies called weak (compact-open) and strong (Whitney) topology. They agree when M is compact. We say that a subset of a topological space is residual if it contains a countable intersection of open dense subsets. The Baire property for C∞(M,N) then guarantees that it is automatically dense. This holds for both topologies but is almost exclusively used for the strong one. Clearly every residual subset of C∞(M,N) for the strong topology is also residual for the weak topology. The following is our main theorem in which we denote by J imm(D,M) the subspace of all jets of immersions. Theorem A. Let D, M , N be manifolds, Y ⊆ J imm(D,M) and Z ⊆ J(D,N) submanifolds. Let us further assume that σY t σZ , where σY = σ|Y : Y ⊆ J(D,M)→ D and σZ = σ|Z : Z ⊆ J(D,N)→ D are the restrictions of the source maps. For a smooth map f : M → N let f∗|Y denote the map Y  // J imm(D,M) f∗ // J(D,N) . Then the subset X := { f ∈ C∞(M,N) ∣∣ f∗|Y t Z} 2000 Mathematics Subject Classification: primary 57R35; secondary 57R45.