Regular maps between Frechet manifolds and pullbacks - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T05:00:20Z http://mathoverflow.net/feeds/question/98160 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/98160/regular-maps-between-frechet-manifolds-and-pullbacks Regular maps between Frechet manifolds and pullbacks David Roberts 2012-05-28T01:50:49Z 2012-05-28T01:50:49Z <p>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 <a href="http://ncatlab.org/nlab/show/Fr%C3%A9chet+manifold" rel="nofollow">Frechet manifolds</a>, 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 <a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.bams/1183549049" rel="nofollow">paper on the Nash-Moser embedding theorem</a>). So we could define a surjective regular map between Frechet manifolds as being a submersion in this sense. My first question is then:</p> <blockquote> <p>What is the full definition of a smooth, regular map of Frechet spaces?</p> </blockquote> <p>'Maximal rank' on each tangent space is not enough, as we have seen.</p> <p>But there are two things flowing on from this which I don't quite get. One is that, considering Frechet manifolds as <a href="http://ncatlab.org/nlab/show/diffeological+space" rel="nofollow">diffeological spaces</a>, there is another concept that is used, namely being a <em>subduction</em> (e.g. <a href="http://mathoverflow.net/questions/48567/advantages-of-diffeological-spaces-over-general-sheaves/49242#49242" rel="nofollow">this MO answer</a>), 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. </p> <blockquote> <p>Can we say anything about whether the pullback diffeological space is a Frechet manifold?</p> </blockquote> <p>The second thing is that really in the case of constructing pullbacks of smooth manifolds, all we need is <a href="http://ncatlab.org/nlab/show/transversal+maps" rel="nofollow">transversality</a> of a pair of smooth maps with common codomain. At <a href="http://en.wikipedia.org/wiki/Transversality_%28mathematics%29" rel="nofollow">this wikipedia page</a>, versions for Banach spaces are mentioned, but no details, and clearly the step from Banach to Frechet spaces is another step removed.</p> <blockquote> <p>What is the correct definition of transversal maps of Frechet manifolds, and what is the statement about existence of pullbacks of transversal maps?</p> </blockquote>