Difference between revisions of "Fundamental group"
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If <math>\circ,\cdot</math> share the same unit <math>e</math> (such that <math>a \cdot e = e \cdot a = a \circ e = e \circ a = a</math>) then <math>\cdot = \circ</math> and both are abelian. | If <math>\circ,\cdot</math> share the same unit <math>e</math> (such that <math>a \cdot e = e \cdot a = a \circ e = e \circ a = a</math>) then <math>\cdot = \circ</math> and both are abelian. | ||
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Revision as of 01:09, 13 December 2009
Perhaps the simplest object of study in algebraic topology is the fundamental group. Let be a path-connected topological space, and let
be any point. Now consider all possible "loops" on
that start and end at
, i.e. all continuous functions
with
. Call this collection
. Now define an equivalence relation
on
by saying that
if there is a continuous function
with
,
, and
. We call
a homotopy. Now define
. That is, we equate any two elements of
which are equivalent under
.
Unsurprisingly, the fundamental group is a group. The identity is the equivalence class containing the map given by
for all
. The inverse of a map
is the map
given by
. We can compose maps as follows:
One can check that this is indeed well-defined.
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.