I am looking for a possible generalization of the standard Trasversality Theorem which roughly says that transverse maps are generic. For example, see the version below:
From page 74, Theorem 2.1 in Hirsch, Differential Topology, GTM 33
Let us modify the step-up to a more general situation. Now assume furthermore that $M,N$ are both the total spaces of fiber bundles over the same base manifold $B$. It is possible that $M,N$ have different fibers over $B$. Everything else is the same as before. Let $C^infty(M,N,B)$ denote the space of smooth $f:Mto N$ making the following diagram commute (under strong or weak topologies, if it matters in this case):
M @>f>> N\
@V V V @VV V\
B @= B
where the vertical maps are the bundle projections. Let $T(M,N,B;A)$ denote the subspace of those $f$ that are transverse to a submanifold $Asubseteq N$.
My question is: Are maps transverse to $A$ generic in any sense among those maps making the diagram commute? For example, is the subset $T(M,N,B;A)$ residual in $C^infty(M,N,B)$? Could anyone guide me to a reference please?
While my intuition might lead (or mislead?) me to believe that transverse maps are always generic, I am not sure if I need to make any additional assumptions on $A$ or on the fiber bundles $Mto B$ and $Nto B$. Notice that if $B$ is a single point, then the new step-up is the same as the old one. So I am indeed asking for a generalization of the usual transversality theorem here.
In a sense, the questions above ask for a “transversality theorem parametrized by $B$“. I am aware that there is a parametric transversality theorem which seems similar in spirit to what I want. But I don’t think it answers my questions above.