Fundamental Group

Fundamental Group2 Path homotopy 1 r 0 0 t 1 g f hrhr g f hrhr x0x0 x1x1 X H

Fundamental Group3 Simply connected space

Fundamental Group4 Path homotopy classes

Fundamental Group5 Proof of Lemma 1.1 Reflexivity 1 r 0 0 t 1 f f x0x0 x1x1 f f X H

Fundamental Group6 Symmetry 1 r 0 0 t 1 g f h 1-r g f x0x0 x1x1 K X

Fundamental Group7 Transitivity 1 1/2 0 0 t 1 g f h g f h x0x0 x1x1 L K X M

Fundamental Group8 Fundamental set

Fundamental Group9 1.2 Naturality

Fundamental Group Theorem

Fundamental Group11 Multiplication in P(X) and in p(X)

Fundamental Group Lemma 1 r 0 0 1/2 1 gf g f x0x0 x1x1 X L f’g’ HK x2x2 f' g'

Fundamental Group Lemma

Fundamental Group Theorem

Fundamental Group Proof (a) p 0 f ½ g ¾ h 1 0 ¼ ½ 1 f g h

Fundamental Group Proof (b) 0 x 0 * ½ f 1 q f r 0 ½ 1 f x 1 *

Fundamental Group Proof (c1) 0 f ½ f 1 u f v 0 ½ 1 f

Fundamental Group Proof (c2) 0 f ½ f 1 u f v 0 ½ 1 f

Fundamental Group Theorem

Fundamental Group20 Summary

Fundamental Group21 Fundamental Group at a Basepoint

Fundamental Group Theorem p 1 (X,x) is a group.

Fundamental Group Theorem The fundamental group have the functorial properties:

Fundamental Group Corollary Any homeomorphism F:X  Y induces an isomorphism F # :p 1 ( X, x )  p 1 ( Y, F ( x )).