I'm pretty sure I knew about some branched coverings; one example that springs to mind is the 2-to-1 map from the torus to the sphere with two branch points. Unless this is a different sort of thing. On Mon, Nov 2, 2020 at 11:22 PM Dan Asimov <dasimov@earthlink.net> wrote:
I think that for compact orientable surfaces M and N, the condition that
π(M) = d π(N)
for some d β Z+ implies that M is a covering space of N. So as Allan was hoping, the converse is true.
Of course there are also lots of maps between surfaces that aren't covering maps. If g β₯ h then there is always a continuous map M_g β> M_h. Many of these are "branched covering" maps, which if the surfaces are given appropriate metrics can be realized as holomorphic maps between Riemann surfaces. I.e., they preserve angles, except at finitely many points where the angle is multiplied by a positive integer > 1. These are called "branched coverings".
Particularly beautiful maps are based on the so-called Weierstrass P-function, which defines a branched covering S^2 β> T^2.
βDan
Allan Wechsler wrote: ----- Oh, Dan, that's perfect. It slots into a place in my brain so nicely that I think I must have known it before and forgotten it. It explains the presence of a 2-to-1 map from the sphere to the projective plane, maps of arbitrary index from the torus to itself, the *absence *of maps between the sphere and pretty much everything else *except* the projective plane.
But there must be all sorts of weird maps from, say, the triple donut (characteristic -4) to the double donut (characteristic -2). Though, come to think of it, your theorem doesn't demonstrate that such maps are always possible, only that sometimes they are clearly impossible. -----
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