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