Section 52: Problem 7 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
Let
be a topological group with operation
and identity element
. Let
denote the set of all loops in
based at
. If
, let us define a loop
by the rule
(a) Show that this operation makes the set
into a group.
(b) Show that this operation induces a group operation
on
.
(c) Show that the two group operations
and
on
are the same. [Hint: Compute
.]
(d) Show that
is abelian.
(a) Since
,
and
are continuous,
is continuous and
, i.e.
. Moreover, since
is associative, so is
, and further
, and the inverse of
is defined by
.
(b)
is well defined, because
is a homotopy between
and
, and it satisfies group properties induced by
.
(c)
because
(d)
.