Algebraic Topology · Problem 33

The Fundamental Group — Allen Hatcher

Mathematics — Topology · covering-spaces

Problem

In Example 1.44 let

d

be the greatest common divisor of

m

and

n

, and let

m' = m/d

and

n' = n/d

. Show that the graph

T_{m,n}/K

consists of

m'

vertices labeled

a

n'

vertices labeled

b

, together with

d

edges joining each

a

vertex to each

b

vertex. Deduce that the subgroup

K \subset G_{m,n}

is free on

dm'n' - m' - n' + 1

generators.