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. On Mon, Nov 2, 2020 at 9:43 PM Dan Asimov <dasimov@earthlink.net> wrote:

If this "d-to-1 map"

h : X —> Y

is a covering map, then the covering space (the domain) will have d times the number of vertices, edges, and faces of the image.

This means

𝜒(X) = d 𝜒(Y).

I presume we're talking about compact orientable surfaces. Since for them,

𝜒 = 2 - 2g,

this tells us that

2 - 2g_X = d (2 - 2g_Y),

or

g_X = d g_Y - d + 1

—Dan

Jim Propp wrote: ----- What is the relation between the genus of X and the genus of Y when there is a d-to-1 map from X to Y? (Assume that around each y in Y we can find a disk whose preimage consists of d disks.) Do we have genus(X) = d genus(Y) ? -----

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