Definition
The fundamental group of a topological space
It is the homotopy equivalence classes of loops. We consider the group multiplication to be composition of loops i.e.
Proof this is a group
The fundamental group follows the group axioms as shown below:
Group operation
Composition of paths is defined as
Inversion
Inversion is defined as
where
Identity element
There exists a constant path
Well defined for equivalence classes
- Composition of paths descends to homotopy classes
- Constant paths are half identity up to homotopy
For a path
from
and
- Inversion is defined in homotopy classes
Pushforward of continuous maps
Let
is a group homomorphism.
Thus, the fundamental group depends (up to isomorphism) only on
Homotopy equivalence
Let