Re: Looking for a surjection or R^2

Denis Feldmann <denis.feldmann.asupprimer@xxxxxxxxxxxxxxxx> writes:

I am looking for a smooth surjection of R^2 onto itself, not bijective,
with jacobian everywhere not zero.
....
Thanks again. Of course, this has many merits, but one of those was to
guide us towards a completely explicit (C-infinite) function

Here's the example with irregular branched coverings that I promised.
*I* think it's already "completely explicit" as I'm about to give it;
but further explicitation can be done (and will be sketched) at the
expense of simplicity.

As before, identify R^2 with C, and let p(z)=z^3-3z. Then p is a
3-sheeted irregular branched cover of C over C, with critical points
1 and -1, and critical values -2 = p(1) = p(-2) and 2 = p(-1) = p(2).
The preimage p^{-1}(R), call it G, is the union of R and two (real)
parabolas, forming a figure approximately like this:

__\___/__
/ \

in which the 2 doublepoints of G are the critical points of p. Note
that C \ G the union of 6 open 2-cells, on each of which the restriction
of p is a diffeomorphism onto either the upper or the lower half-plane;
in fact the restriction of p to the closure of the open 2-cell is a
homeomorphism onto the closed half-plane, which "straightens out" the
right angle(s) on the boundary but is otherwise smooth. Let K be the
closure of the "bottom" 2-cell, that is, the one that contains the
negative imaginary axis, and U its complement. Then p has no critical
points on U, p(U) = C, and p|U is not injective. Since U is manifestly
diffeomorphic to C, we're done, by my standards. To bring the example
up to what *may* be your more exacting standards, we need only write
down a formula for a diffeomorphism of U to C. There's lots of ways
to do that, which however I decline to do at the moment, thanks.

Lee Rudolph
.