# Regular maps between Frechet manifolds and pullbacks

An oft-used example of a regular map of finite-dimensional smooth manifolds is a submersion. We have the well-known result that the pullback of a submersion exists and is a submersion. For Frechet manifolds, one definition of a submersion $f\colon M\to N$ is a map such that every induced map on tangent spaces $T_m M \to T_f(m) N$ is a split map of Frechet spaces. Here surjectivity is not enough. We know that the pullback of a submersion in this sense exists and is a submersion (cf Hamilton's paper on the Nash-Moser embedding theorem). So we could define a surjective regular map between Frechet manifolds as being a submersion in this sense. My first question is then:

What is the full definition of a smooth, regular map of Frechet spaces?

'Maximal rank' on each tangent space is not enough, as we have seen.

But there are two things flowing on from this which I don't quite get. One is that, considering Frechet manifolds as diffeological spaces, there is another concept that is used, namely being a subduction (e.g. this MO answer), which is something like 'is a submersion on each plot'. Now the pullback of a map of Frechet manifolds which is a subduction is not guaranteed to be a Frechet manifold. If it was I would be very happy.

Can we say anything about whether the pullback diffeological space is a Frechet manifold?

The second thing is that really in the case of constructing pullbacks of smooth manifolds, all we need is transversality of a pair of smooth maps with common codomain. At this wikipedia page, versions for Banach spaces are mentioned, but no details, and clearly the step from Banach to Frechet spaces is another step removed.

What is the correct definition of transversal maps of Frechet manifolds, and what is the statement about existence of pullbacks of transversal maps?

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I found this journals.cambridge.org/… for a general definition of transversality, but the paper is very opaque, and doesn't look, after a skim, that it will have what I need. – David Roberts May 28 '12 at 2:00
David, the link in your comment doesn't seem to work. – Dan Ramras May 28 '12 at 5:13
The paper is 'Manifolds of smooth maps II: transversality' by Truong Cong Nghe. This link to the abstract page should work: journals.cambridge.org/action/… – David Roberts May 28 '12 at 7:31