Fundamental group of the circle

Theorem

Let , loop based at .
Then there’s a group isomorphism

Proof

Take the covering map with .

.
Basepoint .
This determines a bijection .

Product law??
with .
for .
by definition

end point of lift of starting at =

Corollary

Fundamental Theorem of Algebra

Theorem

The disc does not retract to .

Proof

By contradiction.
Let be the inclusion.
Suppose is a retraction,
Consider the fundamental groups:

But , so this is a contradiction because

Corollary

Browen’s Fixed Point Theorem