Algebraic topology
Algebraic topology is the study of topology using methods from abstract algebra. In general, given a topological space, we can associate various algebraic objects, such as groups and rings.
Fundamental Groups
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.
Higher Homotopy Groups
Homology and Cohomology
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