Fundamental group
Perhaps the simplest object of study in algebraic topology is the fundamental group.
Let be a based, topological space (that is,
is a topological space, and
is some point in
). Note that some authors will require
to be path-connected. Now consider all possible "loops" on
that start and end at
, i.e. all continuous functions
with
. Call this collection
(the loop space of
). Now define an equivalence relation
on
by saying that
if there is a (based) homotopy between
and
(that is, if there is a continuous function
with
,
, and
). Now let
be the set of equivalence classes of
under
.
Now define a binary operation (called concatenation) on
by
One can check that if
and
then
, and so
induces a well-defined binary operation on
.
One can now check that the operation makes
into a group. The identity element is just the constant loop
, and the inverse of a loop
is just the loop
traversed in the opposite direction (i.e. the loop
). We call
the fundamental group of
.
Note that the fundamental group is not in general abelian. For example, the fundamental group of a figure eight is the free group on two generators, which is not abelian. However, the fundamental group of a circle is , which is abelian.
More generally, if is an h-space, then
is abelian,
for there is a second multiplication on
given by
, which is "compatible" with the concatenation in the following respect:
We say that two binary operations on a
set
are compatible if, for every
,
If share the same unit
(such that
) then
and both are abelian.
Independence from base point
At this point, one might wonder how significant the choice of base point, , was. As it turns out, as long as
is path-connected, the choice of base point is irrelevant to the final group
.
Indeed, pick consider any other base point . As
is path connected, we can find a path
from
to
. Let
be the reverse path from
to
. For any
, define
by
One can now easily check that
is in fact a well-defined map
, and furthermore, that it is a homomorphism. Now we may similarly define the map
by
. One can now easily verify that
is the inverse of
. Thus
is an isomorphism, so
.
Therefore (up to isomorphism), the group is independent of the choice of
. For this reason, we often just write
for the fundamental of
.