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Path Homotopy Α, Β : I fi X HctsL paths of some end points. Α > p Β if F : I I fi X such that p q Β Α F p q Α Β t constant path Ε p HsL = p for all s reverse path Α HsL =ΑH1 - sL c o n c a t e n a t i o n HΑ*ΒLHsL = : ΑH2 sL s ˛ @0, 1 2D ΒH2 s - 1L s ˛ @1 2, 1D Theorem 57 Α : path from p to q Β : path from q to r Γ : path from r to x (i) Α*Ε q > p Α > p Ε p *Α (ii) Α*Α > p Ε p , Α *Α > p Ε q (iii) HΑ*ΒL *Γ > p Α* H Β*ΓL Proof: (i) last time (ii) Α Α Α(0) Ε p Α(1) Α Α ΑHtL =Α H1 - tL FHs, tL = : ΑH2 sL s £ t 2 ΑHtL t 2 £ s £ 1 - t 2 a H2 s - 1L s ‡ 1 - t 2 Printed by Mathematica for Students

herzig/MAT327-lecturenotes21.pdf · 2011-12-011 w [ ] 0 d 1 w [ ] 0 d 1 w [ ] 0 d 1 w [ ] 0 d 1 w [ ] 0 d 1 w [ ] 0 d 1 w [ ] 0 d 1 w [ ] 0 d 1 w [ ] 0 d 1 w [ ] 0 d 1 w [ ] 0 d 1

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