1. Fundamental Group http://cis.k.hosei.ac.jp/~yukita/

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

3. Fundamental Group3 Simply connected space

4. Fundamental Group4 Path homotopy classes

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

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

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

8. Fundamental Group8 Fundamental set

9. Fundamental Group9 1.2 Naturality

10. Fundamental Group10 1.3 Theorem

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

12. Fundamental Group12 1.4 Lemma 1 r 0 0 1/2 1 gf g f x0x0 x1x1 X L f’g’ HK x2x2 f' g'

13. Fundamental Group13 1.5 Lemma

14. Fundamental Group14 1.6 Theorem

15. Fundamental Group15 1.6 Proof (a) p 0 f ½ g ¾ h 1 0 ¼ ½ 1 f g h

16. Fundamental Group16 1.6 Proof (b) 0 x 0 * ½ f 1 q f r 0 ½ 1 f x 1 *

17. Fundamental Group17 1.6 Proof (c1) 0 f ½ f 1 u f v 0 ½ 1 f

18. Fundamental Group18 1.6 Proof (c2) 0 f ½ f 1 u f v 0 ½ 1 f

19. Fundamental Group19 1.7 Theorem

20. Fundamental Group20 Summary

21. Fundamental Group21 Fundamental Group at a Basepoint

22. Fundamental Group22 3.1 Theorem p 1 (X,x) is a group.

23. Fundamental Group23 3.2 Theorem The fundamental group have the functorial properties:

24. Fundamental Group24 3.3 Corollary Any homeomorphism F:X  Y induces an isomorphism F # :p 1 ( X, x )  p 1 ( Y, F ( x )).