Section 54: Problem 6 Solution
Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
Consider the maps
given
and
. (Here we represent
as the set of complex numbers
of absolute value 1.) Compute the induced homomorphisms
,
of the infinite cyclic group
into itself. [Hint: Recall the equation
.]
The group is cyclic, a generator is
, where
, and
where
, i.e.
and
. If we consider isomorphism
between
and
, then corresponding
. Similarly,
, and
.