Algebraic Topology · Problem 2.2.23

Homology — Allen Hatcher

Mathematics — Topology · euler-characteristic

Problem

Show that if the closed orientable surface

M_g

of genus

g

is a covering space of

M_h

, then

g = n(h-1) + 1

for some

n

, namely,

n

is the number of sheets in the covering. [Conversely, if

g = n(h-1)+1

then there is an

n

-sheeted covering

M_g \to M_h

, as we saw in Example 1.41.]