# differential topology – Transversality theorem for maps between fiber bundles

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):
$$require{AMScd}$$
$$begin{CD} M @>f>> N\ @V V V @VV V\ B @= B end{CD}$$
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.

Posted on