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1 ( ) 1. (X, O) X X X O 1.,X O. 2. U, V O U V O 3. O U λ O, λ Λ, λ Λ U λ O. X O (X, O) ( X ) O X X X O (X, O) X n 1 R 1. n R n n (x 1,...,x n ) x =(x 1,...,x n ) x x = x x x 2 n 1

2 R n 2 x, y x y = (x 1 y 1 ) 2 +(x 2 y 2 ) 2 + +(x n y n ) 2 ( y =(y 1,...,y n )) R n R n U p U ɛ>0 R n p ɛ B n (ɛ, p) ={x R n x p <ɛ} U U p U 2. R (1 R 1 ) {x R x > 0} R {x R x 0} 0 2. (X, O) A X O A O A = {A U U O} (A, O A ) (X, O X ) 3. (X, O), (X, O ) f : X X X X O = {f(u) U O} (X, O) (X, O ) (X, O) = (X, O ) X = X 3. R n n B = {x R n x < 1} B n (0, 1) m n R m R n R n R n R n \{0} 2

3 V λ V λ 1: R n R 1 R (M,O) n 1. (Hausdorff ) M 2 p, q p U, q V, U V = U, V 2. M R n R n M V λ O, λ Λ, M = λ Λ V λ n R n 1 4. (1) R n n R n R n (2) U R n U n U p U R n (3) S n = {x R n+1 x =1} n 1 R 1 R 2 ( ) f : R 1 = R 2 R 1 0 R 1 R 1 \{0}, R 2 f(0) R 2 R 2 \{f(0)} R 2 \{f(0)} 1.2 R 1 \{0} R 1 R 2 3

4 R n R n + R n + = {(x 1,...,x n ) R n x n 0} 5. (M,O) n 1. (Hausdorff ) M 2 p, q 2. M R n R n + M n M R n M int M M = M \ int M M int M n M (n 1) 5. (1) R+ n n R n + = {(x 1,...,x n 1, 0) x 1,...,x n 1 R} R n 1 (2) n n D n = {x R n x 1} n D n = S n [0,1] {t R 0 t 1} (X, O) p X q X f : [0, 1] X f(0) = p, f(1) = q 2 f f f :(X, O) (Y,O ) V O f 1 (V ) O 4

5 p = f(0) q = f(1) X f 0 1 [0, 1] 2: 7. (X, O) 2 p, q X p q 6. R n, S n M n M 4 n D n 7. (1) R n (n 1) (2) S n, D n n ( ( 3 4 5

6 1.4 id : R n R n, x x ρ : R n R n ρ(x 1,...,x n 1,x n )=(x 1,...,x n 1, x n ) id, ρ ρ R n 9. f : X Y f X f f(x) Y 10. X Y f g X Y f g t [0, 1] {f t t [0, 1]} f 0 = f, f 1 = g 11. n M i : R n M i i ρ : R n M i i M M i : R n M M M 2 (M,i) M R n, S n 12. (M,i), (Y,j) n f : M Y f i : R n Y g : R n Y 1.5 n M,M M n B M n B M B 6

7 M M B B M M 3: 2 M M. M M. M \ int B M B M \ int B n 1 B, B M M M M 3 (M M )= M M M, M M M M M B B 2 2 M M M M M M B B M M M M M M M M M M n 1. : M M = M M. 7

8 2. : M (M M ) = (M M ) M. 3. : S n M = M. n S n S 1 [0, 1] S 1 [0, 1] 2 {0, 1} S 1,[0, 1] S 1 S 1 = {(x, y) x, y S 1 } S T 2 4 T 2 g g F g 5 F 0 = S 2, F 1 = T 2 RP 2 2 S 2 = {x R 3 x =1} x x 6 RP 2 g g 1 g F g 5 F 1 = RP 2 1( ). 5 8

9 S 1 S 1 = T 2 = S 1 S 1 4: T 2 2 ( ( g : g g g F g (g 0) g F g (g 1) M 1. M S

10 x x x x = = D 2 6: RP 2. RP g... 7: g 2. M M = M 1 M 2 M 1, M 2 S M 1. M 1,...,M r (r 0) M = M 1... M r 2. M = M 1... M r = M 1... M r M 1,...,M r,m 1,...,M r (r, r 0) r = r M 1,...,M r M i = M i, i =1,...,r, 10

11 (1) 2 2 T 2 (2) 2 2 M,M 1,M 2 M M 1 = M M2 M M 1 = M2 4 Heegaard B 3 S 2 V 1 D D D l 8 V 1 T 2 T 2 V 1 g 2 V g n V 1 9 V g g F g V 0 = B 3 g 0 V g g 11

12 l l V 1 D 8: ( l D. l V Heegaard g 0 V g 2 V V V V V V g 1 V V f : V = V x V f(x) V V V V f V V f V V V f V f V M g, V, V, f g : M = V f V (V,V,f,g) M g Heegaard 3. M g 0 M g Heegaard 8. g 0 g V g 9 R 3 R 3 S 3 S 3 V g V V g S 3 = V g V S 3 g Heegaard 12

13 V 1 V 1 V 1 V g 9: 14. M g(m) g(m) Heegaard M 9. 8 g =0 S 3 = V 0 V = B 3 B 3 g(s 3 )=0 B 3 S 3 S M V 1 V, W 10 a, b, D, p, q, E D V V B 3 a V D b V a p, q, E f : V = W M f = V f W f 13

14 b q a D V p E W 10: q p q q p f(a) W p f(a) 11: f a f(a) W. W p, q f(a). f : V = W M f f f a b f a b : a b W a b M f a f(a) p, q a, p, q f(a) p, q f(a) p 2 q 1 11 f(a) p r q s M f L(r, s) r, s 14

15 S2 S1 S 2 [0, 1] S 2 S 1 12: S 2 [0, 1]. S 2 S 1 L(r, s) = L( r, q) r 0 4. (r, s), (r,s ) r, r 0 L(r, s) L(r,s ) r = r s ±s (mod r) ss ±1(modr) 7 4 Reidemeister Reidemeister L(0, ±1) S 2 S 1 12 S 2 S 2 [0, 1] S 2 L(±1,s) s S 3 7 L(r, s) L(r, s) L(r,s ) r = r s s (mod r) ss 1(modr) 15

16 13: 4.4 V g = Fg V g = Fg F g 8 5 Dehn Dehn S 1 S V a, b V D K : V S

17 K(b) K(a) K(V ) 14: K. 16. K : V S 3 f K : V (K(V )) a K(b) b K(a) S 3 K(V ) V V f K K(V ) SK 3 K Dehn Dehn Dehn 10. Dehn 1 5. Dehn Kirby 6 F F F (1) 17

18 = S 2 15: 12 S 2 T 2 16: 24 T 2 (2) (3) 10 M M (1) (2) (3) (4)

19 (1) (2) (3) (4) 17: (1, 4) (2, 3) 18: Pachner (1,4) Pachner (2,3) Pachner 7 Pachner 7 Turaev-Viro 19

431 a s a s a s d a s a 10 d s 11 f a 12 g s 13 a 14 a 15

431 a s a s a s d a s a 10 d s 11 f a 12 g s 13 a 14 a 15 431 a s a s a s d a sa 10ds 11fa 12gs 13a 14a 15 a s d f g h a s d 10f 11g a 12h s 13j a 14k s 15 432 433 10 11 12 13 14 15 10 11 12 13 14 15 434 10 11 12 13 14 15 10 11 12 13 14 15 10 11 12 13 14 15 435