Re: Topology!

W. Dale Hall wrote: 
Suppose such a map exists. Then it can't be surjective on the first
homology group.  So it lifts to a cover. Etc.



Yes, that's what I persisted in not seeing.

To fill in the "etc." part for the OP, the lifting to a cover
X' then provides a map of positive degree to a surface of genus
 > 2. Here's why:

    For a d-fold cover of a surface of genus g, the Euler
    characteristic is

    d*(2-2g) =  2 - 2(d(g-1)+1)

    so the surface has genus d(g-1)+1, or (g=2): d+1.

    Degree is multiplicative wrt composition of maps, so
    if the total degree is 2, one must have a degree 1
    map to a surface of genus 3.

Since the rank of H_1 is 2g, we see that the cover X' has first
Betti number (here, it's 6) greater than that of X (which is 4).
However, for a map of positive degree to exist, the Betti numbers
of the domain must be no less than those of the codomain. This
gives the contradiction.


A small quibble is that one should also rule out the possibility
that the map factors through an infinite sheeted cover.

A nice followup question would be to enumerate all the possibilities
for the degree of a map betweeen a surface of genus g and a surface
of genus h.
.

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