notes / Studies in Real and Complex Analysis / chapter 4: The Fundamental Group and Covering Spaces / From my notebook
N20 The Brouwer Fixed Point Theorem
Theorem N20.1 (Brouwer fixed point theorem). Every continuous has a fixed point.
Theorem N20.2. There is no retraction from to .
(A retraction is a map with , where is the inclusion.)
There is also an algebraic topology proof of the fundamental theorem of algebra over (cool).
A loop in is a map such that . Since is homeomorphic to , we can also think of it as a map . So homotopy classes of maps .
Lemma N20.3. Let be continuous. The following are equivalent:
- is nullhomotopic (homotopic to a constant map);
- extends to a continuous map ;
- is the trivial homomorphism. (Here .)