❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ❑❍❖❆ ❚❖⑩◆ ✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✖ ▲➊ ❚❍➚ ◆❍❯◆● ◆❍➶▼ ❈❒ ❇❷◆ ❈Õ❆ ❑❍➷◆● ●■❆◆ ❚➷ P➷ ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈ ❈❤✉②➯♥ ♥❣➔♥❤ ✿ ❍➐◆❍ ❍➴❈ ❍⑨ ◆❐■✱ ✷✵✶✽ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ❑❍❖❆ ❚❖⑩◆ ✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✖ ▲➊ ❚❍➚ ◆❍❯◆● ◆❍➶▼ ❈❒ ❇❷◆ ❈Õ❆ ❑❍➷◆● ●■❆◆ ❚➷ P➷ ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈ ❈❤✉②➯♥ ♥❣➔♥❤ ✿ ❍➐◆❍ ❍➴❈ ◆❣÷í✐ ữợ ❍⑨ ◆❐■✱ ✷✵✶✽ ▲❮■ ❈❷▼ ❒◆ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ♥➔② ✤÷đ❝ t❤ü❝ ❤✐➺♥ t↕✐ ❦❤♦❛ ❚♦→♥✱ tr÷í♥❣ ✤↕✐ ❤å❝ sữ ữợ sỹ ữợ ❤å❝ ❝õ❛ ❚❙✳ ◆❣✉②➵♥ ❚➜t ❚❤➢♥❣✳ ❚æ✐ ①✐♥ tä ❧á♥❣ ❝↔♠ ì♥ s➙✉ s➢❝ tỵ✐ t❤➛② ◆❣✉②➵♥ ❚➜t ❚❤➢♥❣ ✤➣ ữợ st s tr sốt q tr➻♥❤ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❙ü ❝❤✉②➯♥ ♥❣❤✐➺♣✱ ♥❣❤✐➯♠ tó❝ ✈➔ ♥❤✐➺t t➻♥❤ tr♦♥❣ ự ỳ ữợ ú t ❧➔ t✐➲♥ ✤➲ q✉❛♥ trå♥❣ ❣✐ó♣ tỉ✐ ❝â ✤÷đ❝ ♥❤ú♥❣ ❦➳t q✉↔ tr➻♥❤ ❜➔② tr♦♥❣ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❚æ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ❝❤õ ♥❤✐➺♠ ❦❤♦❛ ❚♦→♥ ✈➔ ❝→❝ t❤➛② ❝æ tr♦♥❣ ❜ë ♠æ♥ ❍➻♥❤ ❤å❝ ♥â✐ r✐➯♥❣ ❝ơ♥❣ ♥❤÷ ❇❛♥ ❣✐→♠ ❤✐➺✉ tr÷í♥❣ ✤↕✐ ❤å❝ s÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷ ♥â✐ ❝❤✉♥❣✱ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ❣✐ó♣ ✤ï tæ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣✳ ❚→❝ ❣✐↔✳ ✶ ▲❮■ ❈❆▼ ✣❖❆◆ ❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ✤➙② ❧➔ ❝æ♥❣ tr ự r tổ ữủ t ữợ sỹ ữợ t t q✉↔ tr➻♥❤ ❜➔② tr♦♥❣ ❦❤â❛ ❧✉➟♥ ❧➔ tr✉♥❣ t❤ü❝ ✈➔ ❝❤÷❛ tø♥❣ ✤÷đ❝ ❝ỉ♥❣ ❜è tr♦♥❣ ❦❤â❛ ❧✉➟♥ ♥➔♦ ❦❤→❝✳ ❚→❝ ❣✐↔✳ ✷ ▼ö❝ ❧ö❝ ▼ð ✤➛✉ ✹ ✶ ◆❤â♠ ỡ ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ◆❤â♠ ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ❑❤æ♥❣ ❣✐❛♥ ♣❤õ ✷✳✶ ❑❤æ♥❣ ❣✐❛♥ ♣❤õ ✈➔ t➼♥❤ ❝❤➜t ✳ ✷✳✶✳✶ ❈→❝ t➼♥❤ ❝❤➜t ♥➙♥❣ ✳ ✳ ✷✳✷ P❤➙♥ ❧♦↕✐ ❦❤æ♥❣ ❣✐❛♥ ♣❤õ ✳ ✳ ✷✳✸ ❇✐➳♥ ✤ê✐ ♣❤õ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ õ ỡ ỵ ❱❛♥ ❑❛♠♣❡♥ ✸✳✶ ✸✳✷ ✸✳✸ ✸✳✹ ✸✳✺ ✸✳✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ◆❤â♠ tü ❞♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚➼❝❤ tü ❞♦ ❝õ❛ ❝→❝ ♥❤â♠✳ ✳ ✳ ✳ ✳ ✳ ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ◆❤â♠ ❝ì ❜↔♥ ❝õ❛ t➼❝❤ ❝❤➟♣ ❝→❝ ✤÷í♥❣ trá♥ ✳ ◆❤â♠ ❝ì ❜↔♥ ❝õ❛ ①✉②➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇➔✐ t➟♣ →♣ ❞ö♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✻ ✼ ✶✶ ✶✶ ✶✶ ✶✷ ✶✸ ✶✹ ✶✻ ✶✻ ✶✼ ✶✼ ✶✽ ✷✵ ✷✶ ▼ð ✤➛✉ ✶✳ ▲Þ ❉❖ ❈❍➴◆ ✣➋ ❚⑨■ ❑❤ỉ♥❣ ❣✐❛♥ tỉ ♣ỉ ❧➔ ♠ët ✤è✐ t÷đ♥❣ ❝ì ❜↔♥ ❝õ❛ ❚♦→♥ ❤å❝ ♥â✐ ❝❤✉♥❣ ✈➔ ❝õ❛ ❍➻♥❤ ❤å❝ tæ ♣æ ♥â✐ r✐➯♥❣✳ ❱✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➼♥❤ ❝❤➜t tæ ♣æ ✈➔ ♣❤➙♥ ❧♦↕✐ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ tæ ♣æ ❧➔ ❝➛♥ t❤✐➳t✳ ❚r♦♥❣ ✤â ✈✐➺❝ ♣❤➙♥ ❧♦↕✐ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ tỉ ♣ỉ ❧➔ ❜➔✐ t♦→♥ ❦❤â✳ ◆❤â♠ ❝ì ❜↔♥ ❧➔ t➟♣ ❤đ♣ ❝→❝ ✈á♥❣ ❤❛② ❝→❝ ✤÷í♥❣ ❝â ❝ò♥❣ ✤✐➸♠ ✤➛✉✱ ✤✐➸♠ ❝✉è✐ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ tæ ♣æ❀ ♥â ❝❤ù❛ ✤ü♥❣ ❝→❝ t❤æ♥❣ t✐♥ ✈➲ ❤➻♥❤ ❞↕♥❣✱ sè ✧❧é t❤õ♥❣✧ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ tỉ ♣ỉ✳ ◆â✐ ❝→❝❤ ❦❤→❝ ♥❤â♠ ❝ì ởt ố tữủ số ự ợ ộ ❦❤ỉ♥❣ ❣✐❛♥ tỉ ♣ỉ✳ ❈❤♦ t❛ ♠ët ❝ỉ♥❣ ❝ư ❝â t❤➸ t➼♥❤ t♦→♥ ✤÷đ❝✳ ❚ø ✤â ❝ơ♥❣ ✤÷❛ r❛ ♠ët ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ ✤➸ ♣❤➙♥ ❧♦↕✐ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ tỉ ♣ỉ✳ ❱✐➺❝ ♥❣❤✐➯♥ ❝ù✉✱ t➻♠ ❤✐➸✉ ✈➲ ♥❤â♠ ❝ì q trồ õ ỵ ❱➻ ✈➟② tæ✐ ❝❤å♥ ❧➔♠ ✤➲ t➔✐ ❝❤♦ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ✷✳ ▼Ö❈ ✣➑❈❍ ◆●❍■➊◆ ❈Ù❯ ◆➯✉ r❛ ❝→❝❤ t➼♥❤ ♥❤â♠ ❝ì ❜↔♥ ♥❤í ❦❤ỉ♥❣ ❣✐❛♥ ♣❤õ✳ ◆➯✉ r❛ ❝→❝❤ t õ ỡ ỵ ✣➮■ ❚×Đ◆● ❱⑨ P❍❸▼ ❱■ ◆●❍■➊◆ ❈Ù❯ ❈→❝ →♥❤ ①↕ ỗ õ ỡ ổ tổ ổ ✹✳ P❍×❒◆● P❍⑩P ◆●❍■➊◆ ❈Ù❯ ❚r❛ ❝ù✉ t➔✐ ❧✐➺✉✱ ✤å❝ ❤✐➸✉ ✈➔ t➼♥❤ t♦→♥ q✉❛ ❝→❝ ✈➼ ❞ư✳ ✺✳ Þ ◆●❍➒❆ ❑❍❖❆ ❍➴❈ ❱⑨ ❚❍Ü❈ ❚■➍◆ ❈Õ❆ ✣➋ ❚⑨■ ▲➔ ♠ët t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤♦ s✐♥❤ ✈✐➯♥ tr♦♥❣ ✈✐➺❝ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❦❤æ♥❣ ❣✐❛♥ tæ ♣æ✱ ♥❤â♠ ❝ì ❜↔♥✳ ✻✳ ❈❻❯ ❚❘Ĩ❈ ❈Õ❆ ▲❯❾◆ ❱❿◆ ✹ ▲✉➟♥ ỗ ữỡ ữỡ tự tr ỗ ♥❤â♠ ❝ì ❜↔♥✱ ❝→❝ ❧♦↕✐ ❦❤ỉ♥❣ ❣✐❛♥ ♣❤õ ✈➔ t➼♥❤ ❝❤➜t ♥❤â♠ ❝ì ❜↔♥ ❝õ❛ ♥â✳ ❈❤÷ì♥❣ ✷✧❑❤ỉ♥❣ ❣✐❛♥ ♣❤õ✧ tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ♣❤õ✱ ❜✐➳♥ ✤ê✐ ♣❤õ ✈➔ ❝→❝❤ t➼♥❤ ♥❤â♠ ❝ì ❜↔♥ q✉❛ ổ ữỡ ỵ tr õ tỹ ỵ ✈➲ ✈✐➺❝ t➼♥❤ ♥❤â♠ ❝ì ❜↔♥✳ ✺ ❈❤÷ì♥❣ ✶ ◆❤â♠ ❝ì ❜↔♥ ❚r♦♥❣ ♠ư❝ ♥➔② t❛ tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ỗ ổ ỗ ❦❤ỉ♥❣ ❣✐❛♥ ❧✐➯♥ t❤ỉ♥❣ ✤÷í♥❣✱ ❦❤ỉ♥❣ ❣✐❛♥ ❝♦ rót ✤÷đ❝✳ ❚➜t ❝↔ ❝→❝ →♥❤ ①↕ tr♦♥❣ ❦❤â❛ ❧✉➟♥ ♥➔② ✤÷đ❝ ❤✐➸✉ ❧➔ →♥❤ ①↕ ❧✐➯♥ tư❝✱ trø ❦❤✐ ✤÷đ❝ ❝❤➾ ró ỗ ỗ X, Y ❤❛✐ ❦❤æ♥❣ ❣✐❛♥ tæ ♣æ✱ I = [0, 1]✳ ❍❛✐ →♥❤ ①↕ f, g : X → Y ✤÷đ❝ ❣å✐ ỗ ợ tỗ t ởt ①↕ F : X × I → Y s❛♦ ❝❤♦✿ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ F (x, 0) = f (x) F (x, 1) = g(x) ❑❤✐ ✤â✱ t❛ ✈✐➳t f ∀x X F ỗ tứ f ✤➳♥ g✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ✭✣÷í♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ tỉ ♣ỉ✮ ❈❤♦ y0 , y1 ∈ Y ✳ ✣÷í♥❣ tr♦♥❣ Y tø y0 ✤➳♥ y1 ❧➔ ♠ët →♥❤ ①↕ f : I → Y ✈ỵ✐✿ g f (0) = y0 f (1) = y1 ✭P❤➨♣ ❤ñ♣ t❤➔♥❤ ❝õ❛ ❝→❝ ✤÷í♥❣✮ ◆➳✉ f : I → X ❧➔ ♠ët ✤÷í♥❣ tø x0 ✤➳♥ x1✱ ✈➔ g : I → X ❧➔ ♠ët ✤÷í♥❣ tø x1 ✤➳♥ x2✱ t➼❝❤ f ∗ g : I → X ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❧➔ ♠ët ữớ f (2s) ợ s 21 (f ∗ g)(s) = g(2s − 1) ✈ỵ✐ 21 ≤ s ≤ tø x0 ✤➳♥ x2✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ ỗ ữớ ữớ f, g : I Y tr Y tứ y0 y1 ỗ ❧✉➙♥ ✤÷í♥❣ ♥➳✉ ❝â ♠ët →♥❤ ①↕ F : I × I → Y (s, 0) = f (s) F (0, t) = y0 ✈ỵ✐✿ FF (s, ∀s ∈ I ✈➔ ∀t ∈ I ✳ 1) = g(s) F (1, t) = y1 ❑❤✐ ✤â t❛ ✈✐➳t f p g F ữủ ởt ỗ ữớ tø f ✤➳♥ g ✳ ✣æ✐ ❦❤✐ t❛ ✈✐➳t f g ✳ ◗✉❛♥ ❤➺ ♥➔② ❧➔ q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣✱ õ ợ tữỡ ữỡ tữỡ ự ữủ ợ ỗ ỵ [f ] ợ ỗ ❧✉➙♥ ❝õ❛ →♥❤ ①↕ ✣à♥❤ ♥❣❤➽❛ ✶✳✹✳ f : X → Y ✶✳✷ ◆❤â♠ ❝ì ❜↔♥ ▼ët ✤÷í♥❣ tr♦♥❣ X tø x0 ✤➳♥ x0 ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✈á♥❣ t↕✐ ✤✐➸♠ ❝ì sð x0✳ ✣➦t π1(X, x0)❧➔ t➟♣ ❤đ♣ ❝→❝ ợ ỗ ữớ ỏ tr X t ✤✐➸♠ ❝ì sð x0✳ ❱ỵ✐ ♠é✐ [f ], [g] ∈ π1(X, x0)✱ t❛ ✤à♥❤ ♥❣❤➽❛✿ ▼➺♥❤ ✤➲ ✶✳✶✳ [f ] ∗ [g] = [f ∗ g] ❑❤✐ ✤â✱ π1(X, x0) ❝ò♥❣ ✈ỵ✐ ♣❤➨♣ t♦→♥ ” ∗ ” ❧➟♣ t❤➔♥❤ ♠ët ♥❤â♠✳ ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ♠å✐ α ❧➔ ♠ët ✈á♥❣ tr♦♥❣ X t x0 ỵ ởt ỏ t↕✐ x0✳ ❈❤ù♥❣ ♠✐♥❤ [1] ❧➔ ♣❤➛♥ tû ✤ì♥ ✈à õ ởt ỗ ữớ ợ [] [1] = [ 1] = [α]✳ ❚÷ì♥❣ tü α ∗ ❧➔ ♠ët ỗ ợ [] [1] = [ ∗ 1] = [α]✳ ❱➟② [1] ❧➔ ♣❤➛♥ tû ✤ì♥ ✈à ❝õ❛ ♥❤â♠✳ ❈❤ù♥❣ ♠✐♥❤ [α−1] ❧➔ ♣❤➛♥ tû ♥❣❤à❝❤ ✤↔♦ ❝õ❛ [α]✳ ❱ỵ✐ ♠å✐ α ❧➔ ♠ët ✈á♥❣ tr♦♥❣ X t↕✐ x0 t❤➻ α−1 ❝ô♥❣ ❧➔ ♠ët ✈á♥❣ t↕✐ x0 s✉② r❛ [α], [α−1] ∈ π1(X, x0)✳ ❱➻ α ởt ỗ ữớ ợ t x0 ♥➯♥ [α] ∗ [α−1 ] = [α ∗ α−1 ] = [1]✳ ❚÷ì♥❣ tü✱ [α−1 ] ∗ [α] = [α−1 ∗ α] = [1]✳ ❙✉② r❛ [α−1] ❧➔ ♣❤➛♥ tû ♥❣❤à❝❤ ✤↔♦ ❝õ❛ [α]✳ ❈❤ù♥❣ ♠✐♥❤ ([α] ∗ [β]) ∗ [γ] = [α] ∗ ([β] ∗ [γ]) ∀[α], [β], [] 1(X, x0) rữợ t ú t ự r ( ) ỗ ữớ ✈ỵ✐ α ∗ (β ∗ γ)✳ ❚❤➟t ✈➟②✱ t❛ ❝â✿ ✼ α(4s) ✈ỵ✐ ≤ s ≤ 1 (α ∗ (β ∗ γ))(s) = β(4s − 1) ✈ỵ✐ ✳ ≤s≤ γ(2s − 1) ✈ỵ✐ ≤ s ≤ α(2s) ✈ỵ✐ ≤ s ≤ ❱➔ ((α ∗ β) ∗ γ)(s) = β(4s − 2) ✈ỵ✐ ≤ s ≤ ✳ γ(4s − 3) ✈ỵ✐ ≤ s ≤ õ õ ởt ỗ F tứ ( ∗ β) ∗ γ ✤➳♥ α ∗ (β ∗ γ) ♠➔ t↕✐ t❤í✐ ❣✐❛♥ t ✤✐ t❤❡♦ ✈á♥❣ α ✈ỵ✐ ≤ s ≤ (1 − t) · 41 + t · 12 = +4 t ❀ s❛✉ ✤â ✤✐ t❤❡♦ ✈á♥❣ β ✈ỵ✐ +4 t ≤ s ≤ (1 − t) · 12 + t · 34 = +4 t ✈➔ ❝✉è✐ ❝ò♥❣ ✤✐ t❤❡♦ ✈á♥❣ γ ✈ỵ✐ +4 t ≤ s ≤ 1✳ 1+t 4s ) ✈ỵ✐ ≤ s ≤ α( 1+t 1+t 2+t ❙✉② r❛ F (s, t) = β(4s − − t) ✈ỵ✐ ≤s≤ 4 γ( 4s − − t ) ✈ỵ✐ + t ≤ s ≤ 2+t ❍❛② ([α] ∗ [β]) ∗ [γ] = [α] ∗ ([β] ∗ [γ])✳ ❙✉② r❛ ” ∗ ” ❝â t➼♥❤ ❝❤➜t ❦➳t ❤ñ♣✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✺✳ π1 (X, x0 ) ❧➔ ♥❤â♠ ❝ì ❜↔♥ ❝õ❛ X ✳ ❱➼ A Rn t ỗ x0 ∈ A t❤➻ π1 (A, x0 ) ❧➔ ♥❤â♠ t➛♠ t❤÷í♥❣✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✻✳ ✭❚➼♥❤ ❤➔♠ tû✮ ❈❤♦ →♥❤ ①↕ h : X → Y ✈ỵ✐ x0 ∈ X ✈➔ y0 = h(x0 )✳ ⑩♥❤ ①↕ h∗ : π1 (X, x0 ) → π1 (Y, y0 ) ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉✿ [f ] → h∗ ([f ]) = [h f ] h ởt ỗ ❝➜✉ ♥❤â♠✳ ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ [f ], [g] ∈ π1(X, x0) t❛ ❝â✿ [f ] ∗ [g] = [f ∗ g] ∈ π1(X, x0)✳ ◆➯♥ h∗([f ∗ g]) = [h ◦ (f ∗ g)] = [h ◦ f ] ∗ [h ◦ g]✳ ◆â✐ ❝→❝❤ ❦❤→❝✿ ▼➺♥❤ ✤➲ ✶✳✷✳ h∗ ([f ] ∗ [g]) = h∗ ([f ]) ∗ h∗ ([g]) ỵ h : (X, x0) → (Y, y0) ✈➔ k : (Y, y0) → (Z, z0) t❤➻ (k ◦ h)∗ = k∗ ◦ h∗✳ ✐✐✮ ◆➳✉ i : (X, x0) → (X, x0) ❧➔ ỗ t t i ỗ ỗ ♥❤➜t ❝õ❛ π1(X, x0) ❈❤ù♥❣ ♠✐♥❤✳ ✐✮ ❚❛ ❝â (k ◦ h) : (X, x0) → (Z, z0) ⇒ (k ◦ h)∗ : π1(X, x0) → π1(Z, z0)✱ [f ] → (k ◦ h)∗ ([f ]) = [k ◦ h ◦ f ]✳ ▼➦t ❦❤→❝✿ h∗ : π1 (X, x0 ) → π1 (Y, y0 ), [f ] → h∗ ([f ]) = [h ◦ f ]✳ k∗ : π1 (Y, y0 ) → π1 (Z, z0 ), [h ◦ f ] → k∗ ([h ◦ f ]) = [k ◦ h ◦ f ]✳ ❙✉② r❛ (k∗ ◦ h∗)([f ]) = (k ◦ h)∗([f ]) ✈ỵ✐ ∀[f ] ∈ π1(X, x0)✳ ✐✐✮ ◆➳✉ i : (X, x0) → (X, x0) ỗ t t õ i : π1 (X, x0 ) → π1 (X, x0 ), [f ] → i∗ ([f ]) = [i ◦ f ] = [f ]✱ ∀[f ] ∈ π1 (X, x0 )✳ ổ ỗ ❦❤ỉ♥❣ ❣✐❛♥ X Y✱ ✈➔ t❛ ♥â✐ ❝❤ó♥❣ t÷ì♥❣ ✤÷ì♥❣ ỗ tỗ t f : X → Y ✈➔ g : Y → X ♠➔ g f idX f g idY ỵ ❤✐➺✉✿ X ≈ Y ▼➺♥❤ ✤➲ ✶✳✸✳ ◆➳✉ X ≈ Y t❤➻ π1 (X, x0 ) ∼ = π1 (Y, y0 ) ✈ỵ✐ x0 ∈ X ✈➔ y0 ∈ Y ❈❤ù♥❣ ♠✐♥❤✳ ❳➨t →♥❤ ①↕ ❝↔♠ s✐♥❤✿ (f g)∗ : π1 (Y, y0 ) → π1 (Y, y0 ) ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤✿ (f g)∗ = idπ1 (Y,y0 ) ✭✈ỵ✐ y0 ∈ Y ♥➔♦ ✤â✮✳ ❱➻ f g ≈ idY tỗ t F : I ì I → Y s❛♦ ❝❤♦✿ F (., 0) = f g F (., 1) = idY ●å✐ [γ] ∈ π1(Y, y0) ✈ỵ✐ γ : I → Y ❧➔ ♠ët G = F0 (γ × id) : I × I → Y t❤ä❛ ♠➣♥✿ ✈á♥❣ t↕✐ y0✳ ❑❤✐ ✤â✱ →♥❤ ①↕ G(t, 0) = F (γ(t), 0) = f g(γ(t)) G(t, 1) = F (γ(t), 1) = γ(t) ✾ ❱➟② f g ◦ γ ≈ γ ✱ tù❝ ❧➔ [f g ◦ γ] = [γ]✳ ❉♦ ✈➟②✿ (f g)∗ = idπ (Y,y )✱ ♥â✐ ❝→❝❤ ❦❤→❝✿ f∗ ◦ g∗ = id✳ ❚÷ì♥❣ tü✿ g∗ ◦ f∗ = id ❱➟② f∗ ❧➔ ✤➥♥❣ ❝➜✉✳ ❚ù❝ ❧➔✿ π1(X, x0) π1(Y, y0) 1(X) = 1(Y ) t X ổ ỗ ợ Y ổ rút ữủ ❈❤♦ A ⊂ X ✳ ❚❛ ♥â✐ X ❝♦ rót ữủ A tỗ t F : X × I → X s❛♦ ❝❤♦✿ ◆❤➟♥ ①➨t ✶✳✶✳ F (x, 0) = x F (x, 1) ∈ A , ∀x ∈ X F (a, t) ∈ A , ∀a ∈ A ❚❛ ❣å✐ F ❧➔ →♥❤ ①↕ ❝♦ rót ❜✐➳♥ ❞↕♥❣ ✤÷❛ X ✈➲ A✳ ❱➼ ❞ư ✶✳✷✳ Rn ❝♦ rót ❜✐➳♥ ❞↕♥❣ ✤÷đ❝ ✈➲ ♠ët ✤✐➸♠✳ ❚❤➟t ✈➟②✿ ❳➨t →♥❤ ①↕ F : Rn × I → Rn , (x, t) → (1 − t)x ▼➺♥❤ ✤➲ ✶✳✹✳ ◆➳✉ X ❝♦ rót ❜✐➳♥ ❞↕♥❣ ✈➲ A t❤➻ X ≈ A ❈❤ù♥❣ ♠✐♥❤✳ ●å✐ F : X × I → X ❧➔ →♥❤ ①↕ ❝â rót ❜✐➳♥ ❞↕♥❣ ✤÷❛ X ✈➲ A✳ ◆➯♥ t❛ ❝â✿f = F (x, 0) : X → X ❧➔ ỗ t g = F (x, 1) : X → A✳ ❳➨t idA : A → X ❧➔ →♥❤ ①↕ ♥❤ó♥❣✳ ❑❤✐ ✤â✿ g ◦ idA : A A ỗ t õ iA ◦ g : X → X ✈➔ F : X × I → X s❛♦ ❝❤♦✿ F (x.0) = idX ✈➔ F (x, 1) ❝â t❤➸ ①❡♠ ♥❤÷ i ◦ g ✳ ❙✉② r❛✿ idA ◦ g ≈ idX ✳ ❱➟② X ≈ A ❍➺ q✉↔ ✶✳✶✳ ◆➳✉ A ❧➔ ❝♦ rót ❜✐➳♥ ❞↕♥❣ ❝õ❛ X t❤➻ π1 (X, a) ≈ π1 (A, a) ❍➺ q✉↔ ✶✳✷✳ π1 (Rn , x) ≈ {1} ▼➺♥❤ ✤➲ ✶✳✺✳ ◆➳✉ X ❧➔ ❧✐➯♥ t❤ỉ♥❣ ✤÷í♥❣ ✈➔ x0, y0 ∈ X t❤➻ π1(X, x0) ∼ = π1 (X, y0 ) ✶✵ ❈❤÷ì♥❣ ✷ ❑❤æ♥❣ ❣✐❛♥ ♣❤õ ✷✳✶ ❑❤æ♥❣ ❣✐❛♥ ♣❤õ ✈➔ t➼♥❤ ❝❤➜t ✣à♥❤ ♥❣❤➽❛ ✷✳✶✳ ❈❤♦ X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ tæ ♣æ✱ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♣❤õ ❝õ❛ X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ X˜ ❝ò♥❣ ✈ỵ✐ →♥❤ ①↕ p : X˜ → X t❤ä❛ ♠➣♥✿ ❱ỵ✐ ✤✐➸♠ x ∈ X ✤➲✉ ❝â ♠ët ❧➙♥ ❝➟♥ ♠ð U ⊂ X s❛♦ ❝❤♦✿ −1 p (U ) = ∪i∈I Ui ✈➔ p|U : Ui U ỗ ổ t è❝✮ ❧➔ ♠➦t ①♦➢♥ ✤✐♥❤ è❝ S ⊂ R3 ❧➔ t➟♣ ❝→❝ ✤✐➸♠ ❝â ❞↕♥❣ (s.cos2πt, s.sin2πt) ✈ỵ✐ (s, t) ∈ (0, ∞) × R✳ ❳➨t ♣❤➨♣ ❝❤✐➳✉ p : S → R2 − {0}✱ (x, y, z) → (x, y)✳ P❤➨♣ ❝❤✐➳✉ ♥➔② ①→❝ ✤à♥❤ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♣❤õ p : S → R2 − {0} ✈➻ ♠é✐ ✤✐➸♠ ❝õ❛ R2 − {0} ✤÷đ❝ ❝❤ù❛ tr♦♥❣ ♠ët ✤➽❛ ♠ð U tr R2 {0} ợ p1(U ) ỗ ♥❤✐➲✉ ✤➽❛ ♠ð rí✐ r↕❝ ✤➳♠ ✤÷đ❝ tr♦♥❣ S ❝❤✐➳✉ ỗ ổ U t ①↕ p : S → S , p(z) = z n ✳ ●å✐ S = {z ∈ C : |z| = 1}❀ s✉② r❛ z = e2iπt✱ ❧➜② z0 ∈ S ⇒ z0 = e2iπt✳ ▼➔ U = {z : z = e2iπt, t ∈ (t − ε, t + ε)} ♥➯♥✿ p−1 (U ) = {e + } ✈ỵ✐ k = 0, n − i 2iπt n ✷✳✶✳✶ 2iπk n ❈→❝ t➼♥❤ ❝❤➜t ♥➙♥❣ ❈❤♦ ❦❤æ♥❣ ❣✐❛♥ ♣❤õ p : X˜ → X ✱ ởt ỗ ft : Y X ởt →♥❤ ①↕ f˜0 : Y → X˜ ♥➙♥❣ f0✱ t❤➻ tỗ t t ỗ f : Y X˜ ❝õ❛ f˜0 ♠➔ ♥➙♥❣ ft✳ ✶✶ ˜ x˜0 ) → π1 (X, x0 ) ✤÷đ❝ s✐♥❤ ❜ð✐ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ♣❤õ ⑩♥❤ ①↕ p∗ : π1(X, ˜ x˜0 ) → (X, x0 ) ❧➔ ✤ì♥ →♥❤✳ ◆❤â♠ ❝♦♥ ↔♥❤ p∗ (π1 ((X, ˜ x˜0 )) tr♦♥❣ p : (X, (X, x0 ) ỗ ợ ỗ ❝õ❛ ❝→❝ ✈á♥❣ tr♦♥❣ X t↕✐ x0 ♥➙♥❣ ❧➯♥ X˜ ❜➢t ✤➛✉ tø x˜0 ❧➔ ❝→❝ ✈á♥❣✳ ˜ x˜0 ) (X, x0 ) ợ ố ữủ tớ ừ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♣❤õ p : (X, ˜ ❧✐➯♥ t❤æ♥❣ ✤÷í♥❣ ❜➡♥❣ ❝❤➾ sè ❝õ❛ p∗ (π1 ((X, ˜ x˜0 )) tr♦♥❣ π1 (X, x0 )✳ X, X ˜ x˜0 ) → (X, x0 ) ✈➔ ♠ët →♥❤ ●✐↔ sû ❝❤♦ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♣❤õ p : (X, ①↕ f : (Y, y0) → (X, x0) ✈ỵ✐ Y ❧✐➯♥ t❤ỉ♥❣ ✤÷í♥❣ ✈➔ ❧✐➯♥ t❤ỉ♥❣ ✤÷í♥❣ ˜ x˜0 ) ❝õ❛ f tỗ t ữỡ t ởt f : (Y, y0)rightarrow(X, ˜ x˜0 )✳ f∗ (π1 (Y, y0 )) ⊂ p∗ (π1 ((X, ❈❤♦ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♣❤õ p : X˜ → X ✈➔ ♠ët →♥❤ ①↕ f : Y → X ✱ ♥➳✉ ❤❛✐ ♥➙♥❣ f˜1, f˜2 : Y → X˜ ❝õ❛ f ❝ò♥❣ t↕✐ ♠ët ✤✐➸♠ ❝õ❛ Y ✈➔ Y ❧✐➯♥ t❤ỉ♥❣ t❤➻ ❝ò♥❣ t↕✐ ♠å✐ ✤✐➸♠ tr➯♥ Y ✳ ✷✳✷ P❤➙♥ ❧♦↕✐ ❦❤æ♥❣ ❣✐❛♥ ♣❤õ ❈❤♦ p1 : X˜1 → X ✈➔ p2 : X˜2 → X ❧➔ ❤❛✐ ❦❤æ♥❣ ❣✐❛♥ ♣❤õ✳ ✣➥♥❣ ❝➜✉ ❣✐ú❛ p1 p2 ữủ ỗ ổ : X˜1 → X˜2 s❛♦ ❝❤♦✿ p2.ϕ = p1✳ ❚❛ ❝á♥ ❣å✐ ϕ ❧➔ ✤➥♥❣ ❝➜✉ ❣✐ú❛ ❤❛✐ ❦❤æ♥❣ ❣✐❛♥ ♣❤õ✳ ▼➺♥❤ ✤➲ ✷✳✶✳ ❈❤♦ X ❧➔ ❧✐➯♥ t❤ỉ♥❣ ✤÷í♥❣✱ ❧✐➯♥ t❤ỉ♥❣ ✤÷í♥❣ ✤à❛ ♣❤÷ì♥❣ ✈➔ ✤ì♥ ❧✐➯♥ ❜→♥ ✤à❛ ♣❤÷ì♥❣✳ ❍❛✐ ❦❤ỉ♥❣ ❣✐❛♥ ♣❤õ ❧✐➯♥ t❤ỉ♥❣ ✤÷í♥❣ p1 : −1 X˜1 → X ✈➔ p2 : X˜2 → X ❀ x˜1 ∈ p−1 (x0 )✱ x˜2 ∈ p2 (x0 ) õ tỗ t ởt f p1 ✈➔ p2 ♠➔ f (x˜1) = x˜2 ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ p1 (π1(X˜1, x˜1)) = ✣à♥❤ ♥❣❤➽❛ ✷✳✷✳ ∗ p2 (1 (X2 , x2 ) ỵ ỵ X ổ tổ ✤÷í♥❣✱ ❧✐➯♥ t❤ỉ♥❣ ✤÷í♥❣ ✤à❛ ♣❤÷ì♥❣ ✈➔ ✤ì♥ ❧✐➯♥ ♥û❛ ữỡ õ tỗ t s ỳ t ❝→❝ ❧ỵ♣ ✤➥♥❣ ❝➜✉ ❜↔♦ t♦➔♥ ✤✐➸♠ ❝ì sð ❝õ❛ ❝→❝ ❦❤æ♥❣ ˜ x˜0 ) → (X, x0 ) ✈➔ t➟♣ ❝→❝ ♥❤â♠ ❝♦♥ ❣✐❛♥ ♣❤õ ❧✐➯♥ t❤ỉ♥❣ ✤÷í♥❣✿ p : (X, ❝õ❛ π1(X, x0)✳ ◆➳✉ ❜ä q✉❛ ✤✐➸♠ ❝ì sð✱ t❤➻ s♦♥❣ →♥❤ ✤â ❝❤♦ ♠ët s♦♥❣ →♥❤ ❣✐ú❛ ❝→❝ ❧ỵ♣ ✤➥♥❣ ❝➜✉ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ♣❤õ ❧✐➯♥ t❤ỉ♥❣ ữớ p : X X ợ ❤ñ♣ ❝õ❛ ❝→❝ ♥❤â♠ ❝♦♥ ❝õ❛ π1(X, x0) ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ✈ỵ✐ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ♣❤õ p : ˜ x˜0 ) → (X, x0 )✱ ✈✐➺❝ t❤❛② ✤ê✐ ✤✐➸♠ ❝ì sð x˜0 ❜➯♥ tr♦♥❣ p−1 (x0 ) t÷ì♥❣ (X, ˜ x0 )) ❜ð✐ ♥❤â♠ ❝♦♥ ❧✐➯♥ ❤đ♣ ❝õ❛ π1 (X, x0 )✳ ù♥❣ ✈ỵ✐ ✈✐➺❝ t❤❛② p∗(π1(X, ✶✷ ●✐↔ sû r➡♥❣ x˜1 ❧➔ ✤✐➸♠ ❝ì sð ❦❤→❝ tr♦♥❣ p−1(x0)✱ ✈➔ ❝❤♦ γ˜ ❧➔ ♠ët ✤÷í♥❣ tø x˜0 ✤➳♥ x1✳ ❚❤➻ γ˜ ❝❤✐➳✉ ♠ët ✈á♥❣ γ ❧➯♥ X ✤↕✐ ❞✐➺♥ ❝❤♦ ♠ët ˜ xi )) ✈ỵ✐ i = 0, 1✳ ♣❤➛♥ tû ♥➔♦ ✤â g ∈ π1(X, x0)✳ ✣➦t Hi = p∗(π1(X, ❚❛ ❝â g−1H0g ⊂ H1 ✈➻ ✈ỵ✐ f˜ ❧➔ ♠ët ✈á♥❣ t↕✐ x˜0 t❤➻ γ˜ · f˜ · γ ❧➔ ♠ët ✈á♥❣ t↕✐ x˜1✳ ❚÷ì♥❣ tü gH1g−1 ⊂ H0✳ ❉♦ ✤â✿ H1 ⊂ g−1H0g✳ ❚ø ❝→❝ ❜❛♦ ❤➔♠ t❤ù❝ tr➯♥ t❛ ✤÷đ❝ ⇒ g−1H0g = H1✳ ❉♦ ✤â✱ t❤❛② ✤ê✐ ❝ì sð tø x˜0 s❛♥❣ x˜1 ❧➔♠ t❤❛② ✤ê✐ H0 s❛♥❣ ♥❤â♠ ❝♦♥ ❧✐➯♥ ❤đ♣ H1 = g−1H0g✳ ◆❣÷đ❝ ❧↕✐✱ ✤➸ t❤❛② ✤ê✐ H0 s❛♥❣ ♠ët ♥❤â♠ ❝♦♥ H1 = g−1H0g✱ ❝❤å♥ ♠ët ✈á♥❣ γ ✤↕✐ ❞✐➺♥ g✱ ♥➙♥❣ ❧➯♥ ✤÷í♥❣ γ˜ ❜➢t ✤➛✉ t↕✐ x˜0✱ ✈➔ ❝❤♦ x˜0 = γ˜✳ ❱ỵ✐ ❧➟♣ ❧✉➟♥ trữợ õ t ữủ H1 = g1H0g ♣❤õ ❱ỵ✐ ❦❤ỉ♥❣ ❣✐❛♥ ♣❤õ p : X˜ → X ✱ ♠ët ♣❤➨♣ ✤➥♥❣ ❝➜✉ ˜ →X ˜ ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❜✐➳♥ ✤ê✐ ♣❤õ✳ X ▼➺♥❤ ✤➲ ✷✳✷✳ ❚➟♣ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❝ò♥❣ ✈ỵ✐ ♣❤➨♣ ❤đ♣ t❤➔♥❤ ❧➟♣ t ởt õ ỵ G(X) ❚➼♥❤ π1 (S ) = ? ❚❛ ❝â (R, exp) tr♦♥❣ ✤â (exp(t) = e2iπt) ❧➔ ♣❤õ ♣❤ê ❞ö♥❣ S ỵ tr t õ G(R) ∼ = π1 (S )✳ ●✐↔ sû✱ h ∈ G(R) t❤➻ exp(h(x)) = exp(x) ❤❛② h(x) = x + n(x) , n(x) ∈ Z s✉② r❛ n(x) = h(x) − x : R → Z ❧➔ →♥❤ ①↕ ❧✐➯♥ tư❝ tr➯♥ t➟♣ rí✐ r↕❝✱ ❤❛② n(x) = const s✉② r❛ h(x) = ˜ = Z x + n , n ∈ Z ♥➔♦ ✤â✳ ❱➟② G(X) ❱➼ ❞ö ✷✳✹✳ ❱ỵ✐ ❦❤ỉ♥❣ ❣✐❛♥ ♣❤õ S → S ✱ z → z n ✱ ❝→❝ ❜✐➳♥ ✤ê✐ ♣❤õ ❧➔ ˜ = Zn ❝→❝ ✈á♥❣ ❧✉➙♥ ♣❤✐➯♥ ❝õ❛ S q✉❛ ❝→❝ ❣â❝ ❜ë✐ ❝õ❛ 2πn ✳ ❱➟② G(X) ❚➼♥❤ ❝❤➜t s❛✉ ❧➔ ❤➺ q✉↔ trü❝ t✐➳♣ ❝õ❛ t➼♥❤ ❝❤➜t ♥➙♥❣ ✤÷í♥❣ t❤ù ♥❤➜t✳ ˜ ❧➔ ❧✐➯♥ t❤ỉ♥❣ ✤÷í♥❣✳ ▼ët ❜✐➳♥ ✤ê✐ ♣❤õ ❤♦➔♥ ▼➺♥❤ ✤➲ ✷✳✸✳ ●✐↔ sû G(X) t♦➔♥ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ↔♥❤ ❝õ❛ ♠ët ✤✐➸♠ ♥➔♦ ✤â✳ ✣➦❝ ❜✐➺t✱ ❝❤➾ ♣❤➨♣ ❜✐➳♥ ˜ ✤ê✐ ỗ t õ t ố ởt X ✣à♥❤ ♥❣❤➽❛ ✷✳✸✳ ✶✸ ✣à♥❤ ♥❣❤➽❛ ✷✳✹✳ ✭❑❤æ♥❣ ❣✐❛♥ ♣❤õ ❝❤✉➞♥ t➢❝✮ ❑❤ỉ♥❣ ❣✐❛♥ ♣❤õ p : X˜ → ✤÷đ❝ ❣å✐ ❧➔ ❝❤✉➞♥ t➢❝ ♥➳✉ ✈ỵ✐ ♠é✐ x ∈ X ✈➔ ♠é✐ ❝➦♣ ♥➙♥❣ x˜, x˜ ❝õ❛ x t❤➻ ❝â ♠ët ❜✐➳♥ ✤ê✐ ♣❤õ ❜✐➳♥ x˜ t❤➔♥❤ x˜ ❱➼ ❞ö ✷✳✺✳ ❈→❝ ❜✐➳♥ ✤ê✐ ♣❤õ tr♦♥❣ ✈➼ ❞ö ✷✳✸ ✈➔ ✷✳✹ ð tr➯♥ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♣❤õ ❝❤✉➞♥ t➢❝✳ X ✷✳✸✳✶ ❚➼♥❤ ♥❤â♠ ❝ì ❜↔♥ ˜ x˜0 ) → (X, x0 ) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♣❤õ ❧✐➯♥ t❤æ♥❣ ỵ p : (X, ữớ ổ ❣✐❛♥ ❧✐➯♥ t❤ỉ♥❣ ✤÷í♥❣✱ ❧✐➯♥ t❤ỉ♥❣ ✤÷í♥❣ ✤à❛ ♣❤÷ì♥❣ X ✱ ˜ x˜0 )) ⊂ π1 (X, x0 )✳ ❑❤✐ ✤â✿ ✈➔ ❝❤♦ H ❧➔ ♥❤â♠ ❝♦♥ p∗(π1(X, ✐✱ ❑❤æ♥❣ ❣✐❛♥ ♣❤õ X˜ ❝❤✉➞♥ t➢❝ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ H ❧➔ ♥❤â♠ ❝♦♥ ❝❤✉➞♥ t➢❝ ❝õ❛ π1(X, x0)✳ ˜ ✤➥♥❣ ợ tữỡ N (H)/H õ N (H) ♥❤â♠ ❝❤✉➞♥ ❤â❛ ✐✐✱ G(X) ❝õ❛ H tr♦♥❣ π1(X, x0) ˜ ✤➥♥❣ ❝➜✉ ✈ỵ✐ π1 (X, x0 )/H ♥➳✉ X ˜ ❧➔ ♣❤õ ❝❤✉➞♥ t➢❝✳ ❉♦ ◆â✐ r✐➯♥❣✱ G(X) ˜ ≈ π1 (X, x0 )✳ ✤â ✈ỵ✐ ♣❤õ ♣❤ê ❞ư♥❣ X˜ → X ❝❤ó♥❣ t❛ ❝â G(X) ❈❤ù♥❣ ♠✐♥❤✳ ❚r♦♥❣ ự ỵ t ❝ì sð x˜0 ∈ p−1(x0) t❤➔♥❤ x˜1 ∈ p−1(x0) t÷ì♥❣ ù♥❣ ✈ỵ✐ ✈✐➺❝ ❧✐➯♥ ❤đ♣ H ❜ð✐ ♠ët ♣❤➛♥ tû [γ] ∈ π1(X, x0) ð ✤â γ ♥➙♥❣ ❧➯♥ ♠ët ✤÷í♥❣ tø γ˜ tø x˜0 ✤➳♥ x˜1 ✳ ❉♦ ✤â✱ [γ] ♥➡♠ tr♦♥❣ ♥❤â♠ ❝❤✉➞♥ ❤â❛ N (H) ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ˜ x˜0 ) = π1 (X, ˜ x˜1 ) tữỡ ữỡ sỹ tỗ t ởt ♣❤õ p∗ (π1 (X, ❜✐➳♥ x˜0 t❤➔♥❤ x˜1✳ ❉♦ ✤â ❦❤æ♥❣ ❣✐❛♥ ♣❤õ ❧➔ ❝❤✉➞♥ t➢❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ N (H) = π1 (X, x0 )✳ ❚ù❝ ❧➔ ❦❤✐ H ❧➔ ♥❤â♠ ❝♦♥ ❝❤✉➞♥ t➢❝ ❝õ❛ π1 (X, x0 )✳ ˜ , [γ] → τ ✈ỵ✐ τ ❧➔ ❜✐➳♥ ✤ê✐ ♣❤õ ❜✐➳♥ x˜0 ❳➨t →♥❤ ①↕ ϕ : N (H) → G(X) t❤➔♥❤ x˜1 tr♦♥❣ ❦❤→✐ ♥✐➺♠ tr➯♥✳ ❚❛ õ ởt ỗ t ❧➔ ♠ët ✈á♥❣ ❦❤→❝ t÷ì♥❣ ù♥❣ ❜✐➳♥ ✤ê✐ ♣❤õ τ ❜✐➳♥ x˜0 t❤➔♥❤ x˜1 t❤➻ γ.γ ♥➙♥❣ ✤➳♥ γ˜.(τ γ˜ )✱ ♠ët ✤÷í♥❣ tø x˜0 s❛♥❣ τ (x˜1) = τ τ (x˜0)✱ ♥➯♥ τ.τ ❧➔ ❜✐➳♥ ✤ê✐ ♣❤õ t÷ì♥❣ ù♥❣ [γ][γ ]✳ ❚❤❡♦ tr➯♥ t❛ ❝â ϕ ❧➔ t♦➔♥ →♥❤✳ ❍↕t ♥❤➙♥ ❝õ❛ ♥â ❝❤ù❛ ❝→❝ ❧ỵ♣ [γ] ♥➙♥❣ ❧➯♥ ❝→❝ ✈á♥❣ tr♦♥❣ X˜ ❤❛② Kerϕ = H ✳ ˜ t q ỵ ỗ t ❝â G(X) N (H)/H ✳ ✶✹ ◆➳✉ t→❝ ✤ë♥❣ ❝õ❛ ♠ët ♥❤â♠ G tr➯♥ ♠ët ❦❤æ♥❣ ❣✐❛♥ Y t❤ä❛ ♠➣♥ ♠é✐ y ∈ Y ❝â ♠ët ❧➙♥ ❝➟♥ U s❛♦ ❝❤♦ ♠å✐ ↔♥❤ g(U ) ✈ỵ✐ ♠å✐ g ∈ G t❤❛② ✤ê✐ ❧➔ rí✐ ♥❤❛✉✱ ♥â✐ ❝→❝❤ ❦❤→❝ g1 (U ) ∩ g2 (U ) = ∅ ⇔ g1 = g2 ✱ t❤➻✿ ✐✱ ⑩♥❤ ①↕ t❤÷ì♥❣ p : Y → Y /G , p(y) = Gy ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♣❤õ ❝❤✉➞♥ t➢❝✳ ✐✐✱ G ❧➔ ♥❤â♠ ❝→❝ ❜✐➳♥ ✤ê✐ ♣❤õ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ♣❤õ Y → Y /G ♥➳✉ Y ❧➔ ❧✐➯♥ t❤ỉ♥❣ ✤÷í♥❣✳ ✐✐✐✱ G ❧➔ ✤➥♥❣ ❝➜✉ ✈ỵ✐ π1(Y /G)/p∗(π1(Y )) ♥➳✉ Y ❧➔ ❧✐➯♥ t❤ỉ♥❣ ✤÷í♥❣✱ ❧✐➯♥ t❤ỉ♥❣ ✤÷í♥❣ ✤à❛ ♣❤÷ì♥❣✳ ❱➼ ❞ư ✷✳✻✳ ❳➨t →♥❤ ①↕ ♣❤õ✿ p : R → S ❚❛ ❝â π1(S 1) ∼ = G(R)✳ ❚❤❡♦ ✈➼ ❞ö tr➯♥ t❛ ❝â✿ G(R ∼ = R)✳ ❱➟② π1(S 1) ∼ = Z✳ ❱➼ ❞ư ✷✳✼✳ ●å✐ T = S1 × S1 ❧➔ ♠ët ①✉②➳♥✳ ●å✐ p : R → S1 ởt t ỵ ❧➔ ♠ët →♥❤ ①↕ p × p : R × R → S1 × S1 ♣❤õ✳ ❚❛ ❝â✿ G(p × p) ∼ = Z × Z✳ ❱➟② π1(S1 × S1) =ZìZ ữỡ ỵ ỵ ởt ữỡ t õ ❝ì ❜↔♥ ❝õ❛ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ t❤ỉ♥❣ q✉❛ ✈✐➺❝ ❝❤✐❛ ❦❤ỉ♥❣ ❣✐❛♥ t❤➔♥❤ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ ♠➔ ♥❤â♠ ❝ì ❜↔♥ ✤➣ ❜✐➳t✳ ❇➡♥❣ ❝→❝❤ sû ❞ö♥❣ ❝â ❤➺ t❤è♥❣ ỵ ú t õ t t õ ỡ ởt ữủ ợ ổ õ tü ❞♦✳ ❈❤♦ S ❧➔ ♠ët t➟♣ ❜➜t ❦ý✳ ●å✐ S −1 ❧➔ ♠ët t➟♣ s♦♥❣ →♥❤ ✈ỵ✐ S ♠➔ tữỡ ự tỷ x S ợ x1 S −1✳ ▼ët tø ✈ỵ✐ ❝❤ú tr♦♥❣ S ❧➔ ♠ët ❞➣② ❤ú✉ ❤↕♥ ❝→❝ ♣❤➛♥ tû ❝õ❛ S ∨ S −1✳ ❉➣② ❦❤ỉ♥❣ ❝â ♣❤➛♥ tû ♥➔♦ ✤÷đ❝ ❣å✐ ❧➔ tø ré♥❣✳ ❈❤♦ ❤❛✐ tø✱ t❛ ❞ü♥❣ ✤÷đ❝ ♠ët tø ♠ỵ✐ ❜➡♥❣ ❝→❝❤ ✤➦t ❤❛✐ tø ✤â ❝↕♥❤ ♥❤❛✉✿ (s1 s2 sn )(t1 t2 tm ) = s1 s2 sn t1 t2 tm t ữợ r ởt tø s1s2 sn ✈ỵ✐ ❝→❝ ❝❤ú tr♦♥❣ S ✱ ♥➳✉ ❤❛✐ tø ❧✐➯♥ t✐➳♣ ❝â ❞↕♥❣ s ✈➔ s−1 t❤➻ t❛ ①â❛ ❜ä✳ ◆❤➟♥ ①➨t ✸✳✶✳ ❚➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ tø ✈➔ ❝❤ú tr♦♥❣ S ✱ ✈ỵ✐ t♦→♥ tû ✤à♥❤ ♥❣❤➽❛ ð tr➯♥ ❧➟♣ t❤➔♥❤ ♠ët ♥❤â♠✳ P❤➛♥ tû ✤ì♥ ✈à ❧➔ tø ré♥❣✱ ✈➔ ♣❤➛♥ tû −1 −1 ♥❣❤à❝❤ ✤↔♦ ❝õ❛ tø s1s2 sn ❧➔ s−1 n s2 s1 õ ỗ tứ ợ ❝❤ú tr♦♥❣ S ✤÷đ❝ ❣å✐ ❧➔ ♥❤â♠ tü ❞♦ s✐♥❤ S ỵ < S > ✸✳✶✳ ◆❤â♠ < {a} > s✐♥❤ ❜ð✐ ♠ët ♣❤➛♥ tû a t ỵ < a > õ t❤➸ ✈✐➳t < a >= {an; n ∈ Z} tr♦♥❣ ✤â am.an = am+n ✈➔ ♣❤➛♥ tû ✶✻ ✤ì♥ ✈à ❧➔ a0✳ ❚❛ ❝â✿ < a >∼ = (Z, +) ✣à♥❤ ♥❣❤➽❛ ✸✳✷✳ ❈❤♦ G ❧➔ ♠ët t➟♣ ✈➔ R ❧➔ t➟♣ ❝→❝ tø ✈ỵ✐ ❝❤ú tr♦♥❣ G✱ tù❝ ❧➔ R ❧➔ t➟♣ ❝♦♥ ❤ú✉ ❤↕♥ ❝õ❛ G✳ ●å✐ N ❧➔ ♥❤â♠ ❝♦♥ ❝❤✉➞♥ t➢❝ ❝õ❛ G ❝❤ù❛ R✳ ❑❤✐ õ õ tữỡ < G > /N ữủ ỵ ❧➔ < G/R >✳ ❈→❝ ♣❤➛♥ tû ❝õ❛ G ✤÷đ❝ ❣å✐ ❧➔ ❝→❝ ♣❤➛♥ tû s✐♥❤ ❝õ❛ ♥❤â♠ ✤â ✈➔ ❝→❝ ♣❤➛♥ tû ❝õ❛ R ✤÷đ❝ ❣å✐ ❧➔ ❝→❝ q✉❛♥ ❤➺✳ ❱➼ ❞ö ✸✳✷✳ < a|a2 >= {a0 , a} ∼ = Z2 ✸✳✷ ❚➼❝❤ tü ❞♦ ❝õ❛ ❝→❝ ♥❤â♠✳ ❈❤♦ G ✈➔ H ❧➔ ❤❛✐ ♥❤â♠✳ ❳➨t t➟♣ t➜t ❝↔ ❝→❝ tø ✈ỵ✐ ❝❤ú tr♦♥❣ G ❤♦➦❝ tr♦♥❣ H ♠➔ tr♦♥❣ ♠ët tø ♥➳✉ ❝→❝ ♣❤➛♥ tû ❧✐➯♥ t tở ũ ởt t ữủ ữợ t♦→♥ tû ♥❤â♠✱ ❝❤➥♥❣ ❤↕♥✿ ba2ab3b−5a = ba3b−2a ◆â✐ r✐➯♥❣✱ ♥➳✉ x ✈➔ x−1 ❦➲ ♥❤❛✉ tr♦♥❣ ♠ët tø t❤➻ ❝❤ó♥❣ tr✐➺t t✐➯✉ ♥❤❛✉✳ ❈→❝ ♣❤➛♥ tû ✤ì♥ ✈à ❝ơ♥❣ ữủ ữợ tứ t ỹ ởt tứ ♠ỵ✐ ❜➡♥❣ ❝→❝❤ ✤➦t ❝❤ó♥❣ ❦➲ ♥❤❛✉✳ ❇ê ✤➲ ✸✳✶✳ ❚➟♣ t➜t ❝↔ ❝→❝ tø ð tr➯♥ ❧➟♣ t❤➔♥❤ ♠ët ♥❤â♠✱ t❛ ❣å✐ ❧➔ t➼❝❤ tü ❞♦ ❝õ❛ G ✈➔ H ỵ G H < g > ∗ < h >:= {g m1 hn1 g m2 hn2 g mk hnk |mi , ni ∈ Z} ỵ U, V X ❧➔ ❝→❝ t➟♣ ♠ð ❧✐➯♥ t❤ỉ♥❣ ✤÷í♥❣ s❛♦ ❝❤♦ U ∩ V ❧✐➯♥ t❤ỉ♥❣ ✤÷í♥❣✱ ❝❤♦ x0 ∈ U ∩ V ✳ ●å✐ iU : U ∩ V → U ✈➔ iV : U ∩ V → V ❧➔ ú õ ỵ (U ∪ V, x0 ) ∼ = π1 (U, x0 ) ∗ π1 (V, x0 ) < {(iU )∗ (α).(iV )−1 ∗ (α)|α ∈ π1 (U ∩ V, x0 )} > ❈❤ù♥❣ ♠✐♥❤✳ ❚r♦♥❣ ♣❤➛♥ ❝❤ù♥❣ ♠✐♥❤ ♥➔② tæ✐ ❝❤➾ tr➻♥❤ ❜➔② ♣❤➛♥ ❝❤ù♥❣ ♠✐♥❤ →♥❤ ①↕ φ : π1(U ), x0 ∗ π1(V, x0) → π1(U ∪ V, x0) ❧➔ t♦➔♥ →♥❤✳ ✶✼ ❈❤♦ γ ❧➔ ♠ët ✈á♥❣ ♥➔♦ ✤â tr♦♥❣ X t↕✐ x0✳ ❚❤❡♦ ❜ê ✤➲ sè ▲❡❜❡s❣✉❡ ❦➳t ❤đ♣ ✈ỵ✐ ♣❤õ ♠ð {γ −1(U ), γ 1(V )} [0, 1] tỗ t ởt ❤♦↕❝❤ ❝õ❛ [0, 1] = t0 < t1 < < tn = s❛♦ ❝❤♦ γ([ti−1, ti] ⊂ U ) ❤♦➦❝ γ −1([ti−1, ti] ⊂ V ) , i = 0, n✱ ✈➔ ❤ì♥ ♥ú❛ ❝❤ó♥❣ t❛ ❝â t❤➸ s➢♣ ①➳♣ ✤➸ γ(ti) ∈ U ∩ V ✳ ✣➦t γi ❧➔ ✤÷í♥❣ γ|[t ,t ] ✤÷đ❝ t❤❛♠ sè ❤â❛ ✤➳♥ ♠✐➲♥ [0, 1]✳❚❤➻ γ ✤÷đ❝ t❤❛♠ sè ❤â❛ ❧➔ γ1 γ2 γn ✳ ❈❤♦ β ❧➔ ♠ët ✤÷í♥❣ tr♦♥❣ U ∩ V tø γ(ti ) ✤➳♥ x0 , i n t ỗ ❧✉➙♥ ✤÷í♥❣ tr♦♥❣ U ∪ V t❛ ❝â✿ i−1 i γ ∼ γ1 γ2 γn −1 ∼ (γ1 β1 ).(β1−1 γ2 β2 ) (βn−1 γn ) ❉♦ ✤â ♠é✐ ✈á♥❣ t x0 tr U V ỗ ỏ t↕✐ x0 tr♦♥❣ U ❤♦➦❝ tr♦♥❣ V ✳ ●å✐ jU : U →→ U ∪ V ✈➔ jV : V U V ữớ ợ ởt t ❝→❝ φ : π1 (U ), x0 ∗ π1 (V, x0 ) → π1 (U ∪ V, x0 ) a1 b1 an bn → (jU )∗ (a1 )(jV )∗ (b1 ) (jU ) (an )(jV ) (bn ) ởt ỗ ❝➜✉✱ ❤❛② φ ❧➔ t♦➔♥ →♥❤✳ ❍✐➸♥ ♥❤✐➯♥ φ ❧➔ ❍➺ q✉↔ ✸✳✶✳ ◆➳✉ U ∩V ✤ì♥ ❧✐➯♥ t❤➻ π1 (U ∩V, x0 ) ∼ = π1 (U, x0 )∗π1 (V, x0 ) ✸✳✹ ◆❤â♠ ❝ì ❜↔♥ ❝õ❛ t➼❝❤ ❝❤➟♣ ❝→❝ ✤÷í♥❣ trá♥ ✣à♥❤ ♥❣❤➽❛ ✸✳✸✳ ❈❤♦ X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❍❛✉s❞♦r❢❢ s❛♦ ❝❤♦✿ n X= Si i=1 ♠➔ Si ỗ ổ ợ ữớ trỏ S sỷ tỗ t↕✐ p ∈ X ♠➔ Si ∩Sj = {p} ♥➳✉ i = j ✳ ❑❤✐ ✤â✿ X ✤÷đ❝ ❣å✐ ❧➔ t ữớ trỏ ỵ X ❧➔ t➼❝❤ ❝❤➟♣ ❝õ❛ ❝→❝ ✤÷í♥❣ trá♥ S1 , , Sn ✳ ●å✐ p ❧➔ ✤✐➸♠ ❝❤✉♥❣ ❝õ❛ ❝→❝ ✤÷í♥❣ trá♥✳ ❑❤✐ ✤â✱ π1(X, p) ❧➔ ♥❤â♠ tü ❞♦✳ ◆➳✉ fi ❧➔ ❝→❝ ✈á♥❣ tr♦♥❣ Si t❤➻ f1 , , fn ❧➔ ❝→❝ ♣❤➛♥ tû s✐♥❤ ❝õ❛ π1 (X, p) ✣➸ ❝❤ù♥❣ ♠✐♥❤ t❛ ❝➛♥ ❜ê ✤➲ s❛✉✿ ✶✽ ❈❤♦ X = A ∪ B ✈ỵ✐ A, B ❧➔ ❝→❝ t➟♣ ❝♦♥ ✤â♥❣ ❝õ❛ X ✳ ❈❤♦ f : A → Y ✈➔ g : B → Y ❧➔ ❝→❝ →♥❤ ①↕ ❧✐➯♥ tö❝✱ s❛♦ ❝❤♦✿ f (x) = g(x) ✈ỵ✐ x ∈ A ∩ B ✳ ❑❤✐ ✤â✱ tỗ t h:XY s h(x) = f (x) ♥➳✉ x ∈ A ✈➔ h(x) = g(x) ♥➳✉ x ∈ B ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝➛♥ ❦✐➸♠ tr❛ →♥❤ ①↕ h : X → Y s❛♦ ❝❤♦✿ x → f (x) ♥➳✉ x ∈ A ✈➔ x → g(x) ♥➳✉ x ∈ B ❧➔ →♥❤ ①↕ ❧✐➯♥ tö❝✳ ❈❤♦ U ⊂ Y ❧➔ t➟♣ ✤â♥❣ t❤➻ h−1(U ) = f −1(U ) ∪ g−1(U )✳ ❱➻ f ❧✐➯♥ tö❝ ♥➯♥ f −1(U ) ✤â♥❣ tr♦♥❣ A✱ ❞♦ ✤â ✤â♥❣ tr♦♥❣ X ✳ ❚÷ì♥❣ tü✱ ✈➻ g ❧✐➯♥ tö❝ ♥➯♥ g−1(U ) ✤â♥❣ tr♦♥❣ B ✱ ❞♦ ✤â ✤â♥❣ tr♦♥❣ X ✳ ❱➟② h−1(U ) = f −1(U ) ∪ g−1(U ) ✤â♥❣ tr♦♥❣ X ✳ ❙✉② r h tử ự ỵ s➩ ❝❤ù♥❣ ♠✐♥❤ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ q✉② ♥↕♣ t❤❡♦ n✳ ❱ỵ✐ n = ❤✐➸♥ ♥❤✐➯♥ ✤ó♥❣✳ ❈❤♦ X ❧➔ t n ữớ trỏ S1, , Sn ợ ✤✐➸♠ ❝❤✉♥❣ p✳ ❚r➯♥ ♠é✐ Si ❧➜② ♠ët ✤✐➸♠ qi = p t wi = Si qi ỵ U = S1 ∪ w2 ∪ ∪ wn ✈➔ V = w1 ∪ S2 ∪ ∪ Sn✳ ❑❤✐ ✤â✿ U ∩ V = w1 ∪ w2 ∪ wn õ wi ỗ ổ ợ ❞♦ ✤â ❝♦ rót ❜✐➳♥ ❞↕♥❣ ✈➲ p✳ ●å✐ Fi : wi × I → wi ❧➔ ❝→❝ →♥❤ ①↕ ❝♦ rót ❜✐➳♥ ❞↕♥❣✳ ❚❤❡♦ ❜ê ✤➲ tr➯♥✱ t❛ ❝â →♥❤ ①↕ ❝♦ rót ❜✐➳♥ ❞↕♥❣ F : U ∩ V × I → U ∩ V ❧➔ ♣❤➨♣ ❝♦ rót ❜✐➳♥ ❞↕♥❣ ✤÷❛ U ∩ V ✈➲ p✳ ❱➟② U V ỡ ỵ ❑❛♠♣❡♥ t❛ ❝â✿ π1 (X, p) ∼ = π1 (U, p) ∗ π1 (V, p) ❚÷ì♥❣ tü ♥❤÷ tr➯♥ t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ U ❝♦ rót ❜✐➳♥ ❞↕♥❣ ✈➲ S ✈➔ V ❝â rót ❜✐➳♥ ❞↕♥❣ ✈➲ t➼❝❤ ❝❤➟♣ ❝õ❛ (n − 1) ✤÷í♥❣ trá♥✳ ❱➟② π1(X, p) ∼ = π1 (U, p) ∗ π1 (V, p)✳ ❚❤❡♦ ❝❤ù♥❣ ♠✐♥❤ q✉② ♥↕♣ t❛ ❝â ✤÷đ❝ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ✶✾ ✸✳✺ ◆❤â♠ ❝ì ❜↔♥ ❝õ❛ ①✉②➳♥ ❚r♦♥❣ ✈➼ ❞ư ✷✳✼✱ t❛ ✤➣ sû ❞ư♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♣❤õ ✤➸ t➼♥❤ ữủ 1(S ì S 1) = Z ì Z✳ ❚r♦♥❣ ♠ư❝ ♥➔② t❛ s➩ ❜✐➸✉ ❞✐➵♥ ♥❤â♠ ❝ì ❜↔♥ ❝õ❛ ①✉②➳♥ t❤æ♥❣ q✉❛ t➼❝❤ tü ❞♦✳ ❚ø ✈➼ ❞ư ✷✳✼✱ t❛ ❝â →♥❤ ①↕ p × p : R × R → S × S 1✳ ❳➨t →♥❤ ①↕ ❤↕♥ ❝❤➳ ❝õ❛ →♥❤ ①↕ ✤â ❧➯♥ I t ữủ = p ì p : I → T = S × S ⑩♥❤ ①↕ ✤â ❜✐➳♥ ❜✐➯♥ ❝õ❛ I ✈➔♦ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ A = S ×{1}∪{1}×S 1❀ A ❝❤➼♥❤ ❧➔ t➼❝❤ ❝❤➟♣ ❝õ❛ ❤❛✐ ✤÷í♥❣ trá♥✳ ⑩♥❤ ①↕ π ❜✐➳♥ ♣❤➛♥ tr♦♥❣ ❝õ❛ I s♦❛♥❣ →♥❤ ❧➯♥ T − A✳ ❱➟② ①✉②➳♥ T ❝â t❤➸ ①❡♠ ❧➔ ❤➻♥❤ ✈✉æ♥❣ I ♠➔ ❞→♥ ❤❛✐ ❝↕♥❤ ❝õ❛ ♥â ❧➯♥ A✳ ỵ s ỵ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍❛✉s❞♦r❢❢✳ A ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ✤â♥❣✱ ❧✐➯♥ t❤ỉ♥❣ ✤÷í♥❣ ❝õ❛ X ✳ ●✐↔ sû ❝â ♠ët →♥❤ ①↕ ❧✐➯♥ tö❝ h : B → X ♠➔ ❜✐➳♥ ♠✐➲♥ tr♦♥❣ B s♦♥❣ →♥❤ ❧➯♥ X − A ✈➔ ❜✐➳♥ S = ∂B ❧➯♥ A✳ ❈❤♦ p ∈ S 1, a = h(p) ❣å✐✿ k : (S , p) → (A, a) ❧➔ h õ ỗ i∗ : π1 (A, a) → π1 (X, a) ❝↔♠ s✐♥❤ tø →♥❤ ①↕ ♥❤ó♥❣ ❧➔ t♦➔♥ →♥❤ ✈➔ ❤↕t ♥❤➙♥ ❧➔ ♥❤â♠ ❝♦♥ ❝❤✉➞♥ t➢❝ ♥❤ä ♥❤➜t ❝õ❛ π1(X, a) ♠➔ ❝❤ù❛ ↔♥❤ ❝õ❛ k∗ : π1(S 1, a) 1(A, a) ỵ õ ỡ ①✉②➳♥✮✳ ◆❤â♠ ❝ì ❜↔♥ ❝õ❛ ①✉②➳♥ S × S õ ỗ tỷ s , β ✈ỵ✐ q✉❛♥ ❤➺ αβα−1β −1 ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ ①✉②➳♥ X = S × S ✈➔ →♥❤ ①↕ h : I → X ❧➔ →♥❤ ①↕ ❤↕♥ ❝❤➳ ❝õ❛ →♥❤ ①↕ ♣❤õ p × p : R × R → S × S 1✳ ❈❤♦ p(0, 0) ∈ BdI 2✳ ❈❤♦ a = h(p) ✈➔ A = h(BdI 2) t tt ỵ tr ✤÷đ❝ t❤ä❛ ♠➣♥✳ ✷✵ ❑❤ỉ♥❣ ❣✐❛♥ A ❧➔ t➼❝❤ ❝❤➟♣ ❝õ❛ ❤❛✐ ✤÷í♥❣ trá♥✱ ♥➯♥ ♥❤â♠ ❝ì ❜↔♥ ❝õ❛ A ❧➔ ♥❤â♠ tü ❞♦✳ ❚❤➟t ✈➟②✱ ♥➳✉ ❝❤ó♥❣ t❛ ✤➦t a0 ❧➔ ♠ët ✤÷í♥❣ a0(t) = (t, 0) ✈➔ b0 ❧➔ ♠ët ✤÷í♥❣ b0(t) = (0, t) ∈ BdI 2✱ t❤➻ ❝→❝ ✤÷í♥❣ α = h ◦ a0 ✈➔ β = h ◦ b0 ❧➔ ❝→❝ ✈á♥❣ tr♦♥❣ A s❛♦ ❝❤♦ [α], [β] ❤➻♥❤ t❤➔♥❤ ❤➺ ❝→❝ ♣❤➛♥ tû s✐♥❤ tü ❞♦ ❝❤♦ π1(A, a)✳ ❈❤♦ a1 ❧➔ ♠ët ✤÷í♥❣ a1(t) = (t, 1) ✈➔ b1 ❧➔ ♠ët ✤÷í♥❣ b1(t) = (1, t) ∈ BdI ✳ ❳➨t ✈á♥❣ f tr♦♥❣ BdI ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ♣❤➨♣ t♦→♥✿ f = a0 ∗ (b1 ∗ (a ∗ b)) t❤➻ f ✤↕✐ ❞✐➺♥ ❝❤♦ ♠ët ♣❤➛♥ tû s✐♥❤ ❝õ❛ π1(BdI 2, p)❀ ✈➔ ✈á♥❣ g = h ◦ f ❜➡♥❣ t➼❝❤ ( ( )) ỵ tr➯♥✱ t❛ ❝â π1(X, a) ❧➔ t❤÷ì♥❣ ❝õ❛ ♥❤â♠ tü ❞♦ s✐♥❤ ❜ð✐ ❤❛✐ ♣❤➛♥ tû tü ❞♦ [α] ✈➔ [β] ✈➔ ♥❤â♠ ❝♦♥ ❝❤✉➞♥ t➢❝ ♥❤ä ♥❤➜t ❝❤ù❛ ♣❤➛♥ tû [α][β][α−1][β −1] ✸✳✻ ❇➔✐ t➟♣ →♣ ❞ö♥❣ ✶✳ ❈❤♦ X ⊂ R3 π1 (R3 − X) ❧➔ ❤ñ♣ ❝õ❛ n ✤÷í♥❣ t❤➥♥❣ ✤✐ q✉❛ ❣è❝ tå❛ ✤ë✳ ❚➼♥❤ ✷✳ ❚➼♥❤ ♥❤â♠ ❝ì ❜↔♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ t❤✉ ✤÷đ❝ tø ❤❛✐ ❤➻♥❤ ♣❤ä♥❣ ①✉②➳♥ S × S ❜➡♥❣ ỗ t ởt ữớ trỏ S ì {x0 } tr ởt ỏ ợ ữớ trỏ S × {x0} t÷ì♥❣ ù♥❣ tr♦♥❣ ♠ët ✈á♥❣ ①✉②➳♥ ❦❤→❝✳ ✸✳ ❈❤➾ r❛ r➡♥❣ π1(R2 − Q2) ❧➔ ❦❤ỉ♥❣ ✤➳♠ ✤÷đ❝✳ ✹✳ ❈❤➾ r❛ r➡♥❣ ❤ñ♣ X ∗ Y ❝õ❛ ❤❛✐ ❦❤ỉ♥❣ ❣✐❛♥ ❦❤→❝ ré♥❣ X ✈➔ Y ❧➔ ✤ì♥ ❧✐➯♥ ♥➳✉ X ❧➔ ❧✐➯♥ t❤ỉ♥❣ ✤÷í♥❣✳ ✺✳ ❈❤➾ r❛ r➡♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ R3 ❧➔ ❤đ♣ ❝→❝ ❤➻♥❤ ❝➛✉ ❝â ❜→♥ ❦➼♥❤ 1 n ✈➔ t➙♠ ( n , 0, 0) ✈ỵ✐ n = 1, 2, ❧➔ ✤ì♥ ❧✐➯♥✳ ✷✶ ❑➌❚ ▲❯❾◆ ❑❤â❛ ❧✉➟♥ ♥❣❤✐➯♥ ❝ù✉ ♥❤â♠ ❝ì ❜↔♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❚ỉ ♣ỉ✳ ❑➳t q✉↔ ✤↕t ữủ ỗ ữ r ữủ ởt số ỡ tr ổ tổ ổ ỗ ổ ỗ õ ỡ ❦❤ỉ♥❣ ❣✐❛♥ ♣❤õ ✈➔ t➼♥❤ ❝❤➜t ♥❤â♠ ❝ì ❜↔♥ ❝õ❛ ♥â✳ ✷✳ ✣÷❛ r❛ ❝→❝❤ t➼♥❤ ♥❤â♠ ❝ì ❜↔♥ q✉❛ ❦❤ỉ♥❣ ❣✐❛♥ ♣❤õ ✈➔ ❝â ✈➼ ❞ư✱ ❜➔✐ t➟♣ ❝ư t❤➸✳ ✸✳ ✣÷❛ r❛ ❝→❝❤ t➼♥❤ ♥❤â♠ ❝ì ❜↔♥ q✉❛ ỵ õ õ ❝ì ❜↔♥ ❝õ❛ t➼❝❤ ❝❤➟♣ ❝→❝ ✤÷í♥❣ trá♥✱ ♥❤â♠ ❝ì ❜↔♥ ❝õ❛ ①✉②➳♥ ✈➔ ❝â ❜➔✐ t➟♣ ❝ö t❤➸✳ ✷✷ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❆✳❍❛t❝❤❡r✱ ✷✵✵✷✱ ❆❧❣❡❜r❛✐❝ t♦♣♦❧♦❣② ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✳ ❬✷❪ ❏✳ ▼✉♥❦r❡s✱ ✶✾✾✾✱ ❚♦♣♦❧♦❣②✭s❡❝♦♥❞ ❡❞✐t✐♦♥✮ P❡❛rs♦♥✳ ✷✸