II R n k +1 v 0,, v k k v 1 v 0,, v k v v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ k σ dimσ = k 1.3. k

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1 II R n k +1 v 0,, v k k v 1 v 0,, v k v v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ kσ dimσ = k 1.3. k σ {v 0,...,v k } {v i0,...,v il } l σ τ < τ τ σ 1.4. R n K 1. σ K τ < σ τ K 2. σ 1, σ 2 K σ 1 σ 2 σ 1 σ K l β l (K) K χ(k) χ(k) = l 0( 1) l β l (K) 1.6. R n K R n K = σ σ K K 1

2 1. R n v 0,, v k (a) v 0,, v k R n (b) k +1 a 0,, a k k i=0 a iv i = 0 k i=0 a i = 0 a 0 = = a k = 0 (c) l k v i v l 0 i k, i l 2. v 0 v m v 0,, v m R n A x, y A 3. m σ = v 0 v m K(σ) L(σ) = K(σ) {σ} L(σ){σ} K(σ) L(σ) 4. i j v i v j (a) K = { v 0, v 1, v 2, v 3, v 4, v 0 v 1, v 0 v 2, v 1 v 2, v 0 v 3, v 0 v 4, v 3 v 4, v 0 v 1 v 2 } (b) K = { v 0, v 1, v 2, v 3, v 4, v 5, v 0 v 1, v 0 v 2, v 1 v 2, v 0 v 3, v 3 v 4, v 3 v 5, v 4 v 5 } ( I ) 5. f 0 : A Y f 1 : B Y X Y A B xf 0 (x) = f 1 (x) f 0 (x), x A f(x) = f 1 (x), x B f: A B Y 6. f: X Y (a) f (b) f X Y (c) f X Y 7. X Y 2

3 K, K 12 K K 1. K = K 2. K K 2.2. k σ = v 0 v 1 v k σ b(σ) = 1 k +1 v k +1 v k +1 v k 2.3. K σ 0 < σ 1 < σ k b(σ 0 ),b(σ 1 ),,b(σ k ) b(σ 0 )b(σ 1 ) b(σ k ) K K K SdK 2.4. K K χ(k ) = χ(k) 2.5. K L K L L K L KL K = L K = L K K L L KL 2.6. φ: K L K j a jv j v 0 v k K j a jφ(v j ) φ( v 0 v k ) φ : K L 2.7. φ: K L φ : K L φ φ 2.8. KL K L 2.9. KL χ(k) = χ(l) KLR n K L = K L K L K L χ(k L) = χ(k)+χ(l) χ(k L) K L = K L K K L L K L = K L χ(k L) = χ(k)+χ(l) χ(k L ) 3

4 1. SdKK G G V(G), E(G) F(G) G 2 V(G) E(G)+F(G) = 2 5. P I(P)P B(P) P = I(P)+ 1 2 B(P) 1 1/2 4

5 X K X f: K X KX 3.2. k σ K(σ) L(σ) = K(σ) {σ} k B k k 1 S k X e X ϕ: (I k, I k ) (e, e) ex ϕe ke dime = k I = [0,1] I k = I k IntI k e = e e ϕ I k I k = IntI k IntI k e 3.4. X D = {e j j J} X DX 1. j k e j e k = 2. X = j J e j 3. dime j = k e j = e j e j k 1 D X 3.5. K K {Intσ σ K} 3.6. XD = {e j j J} J k = {j J dime j k} X Xk X k = j J k e j 3.7. X k X k X k 1 k I k X k + 1 e j (j J k+1 J k ) ϕ j : (I k, I k ) (e j, e j ) X k = X k 1 I k (J k J k 1 )/ (t,j) ϕ j (t) (t I k, j J k J k 1 ) 3.8. XD = {e j j J} χ(x) = k ( 1)k β k β k Xk 5

6 1. (a) f: X Y X x, y x y f(x) = f(y) (b) X X/ p: X f = g p g: X/ Y (c) f g 2. (a) C S 1 = {z C z = 1} (b) x y y x Z R R/Z (c) [0,1] 1 [0,1]/{0,1} 3. I = [0,1] I n n D n = {x R n x 1} 4. X A Y φ: A Y X Y a Af(a) Y X φ Y φ Y X X Y X φ Y p (a) f A = g φ f: X Z g: Y Z h p(x) = f(x) (x X)h p(y) = g(y) (y Y) h: X φ Y Z (b) ex φ: (I n, I n ) (ē, e) X (X e) φ I n I n 6

7 Xn 1. X 2. X R n 4.2. X U = {U j j J} U j n ϕ j : U j V j R n Xn (U j,ϕ j ) U 4.3. n 1. n R n 2. n S n = {x R n+1 x = 1} 3. n RP n = R n+1 {0}/ x = ay a x y D 2 S 1 2n a, a a, a 1 2n S( ) 4.5. n = 1, 2 S(aa 1 ) = S(aa) = S(aba 1 b 1 ) = S(aba 1 b) = S(abb 1 a 1 )S(abab)S(aabb) 7

8 1. X {(x,x) x X} X X 2. f g X Y {x X f(x) = g(x)}x 3. λ Λ X λ X λ p λ f: X λ Λ X λ p λ f: X X λ 4. X = λ Λ X λ (a) X λ X (b) X X λ 5. Xm Y n X Y m+n 6. [0,1] [0,1] 2 (x,0) (x,1), (0,y) (1,y), 0 x, y 1 (a) (b) S 1 S 1 8

9 g T(g), P(g) T(g) = S(a 1 b 1 a 1 1 b 1 1 a g b g a 1 g b 1 g ) P(g) = S(a 1 a 1 a g a g ) T(0) = S χ(t(g)) = 2 2gχ(P(g)) = 2 g 5.3. l S(l) g T(g) P(g) 5.4. D 1, D 2 S 1, S 2 ϕ: C 1 C 2 S 1 IntD 1 S 2 IntD 2 S 1 IntD 1 x D 1 S 2 IntD 2 ϕ(x) D 2 S 1 S 2 S 1 S D 1, D 2 ϕ: D 1 = D g 1 T(g) = T(1) T(1)P(g) = P(1) P(1) g 5.7. S(aba 1 b) = P(2) 5.8 ( ). S 2 T(g) P(g) = 1, 2, ) 5.9. T(g) (g 0) P(g) (g > 0) g 9

10 1. T(p) P(q) 2. S(aba 1 b) = P(2) 3. S(aba 1 b 1 cc) = S(aab b cc) T(1) P(1) = P(1) P(1) P(1) 10

11 ObC hom C (a,b) (a, b C) hom C (b,c) hom C (a,b) hom C (a,c), (f,g) g f 1 a hom C (a,a) C 1. f hom C (a,b) f 1 a = f = 1 b f 2. f hom C (a,b), g hom C (b,c), h hom C (c,d) h (g f) = (h g) f 6.2. Set Top Top 2 Mod Mod 6.3. C f: a b D Ff: Fa Fb hom C (a,b) hom D (Fa,Fb) F F1 a = 1 Fa, F(g f) = Fg Ff F C D F: C D 6.4. α, β: Top 2 Top, α(x,a) = A, β(x,a) = X σ: Mod Mod, σ{a n } = {(σa) n }, (σa) n = A n F, G C D C f: a b D φ = {φ a : Fa Ga a ObC}F G φ: F G Ff Fa Fb φ a Ga Gf φ b Gb 6.6. α, β: Top 2 Top φ: α β, φ (X,A) : A X 11

12 6.7. X A 1, A n 1 (X,A 1,...,A n 1 ) n f(a j ) B j f: X Y n f: (X,A 1,...,A n 1 ) (Y,B 1,...,B n 1 ) n f, g: (X,A 1,...,A n 1 ) (Y,B 1,...,B n 1 ) F(x,0) = f(x), F(x,1) = g(x), x X n F: (X,A 1,...,A n 1 ) I (Y,B 1,...,B n 1 ) f g f 6.8. H: (X,A) {H n (X,A) n Z} : H n (X,A) H n 1 (A) H n (X, )H n (X) 1. f g: (X,A) (Y,B) f = g : H n (X,A) H n (Y,B) 2. H n (A) i H n (X) j H n (X,A) H n 1 (A) i i, j i: (A, ) (X, ) j: (X, ) (X,A) 3. U IntA H n (X U,A U) = H n (X,A) 4. P n 0 H n (P) = 0 H n (X,A) (X,A)n P 0 G = H 0 (P) G G = Z H n (X,A)n 6.9. f: (X,A) (Y,B) g: (X,A) g f 1 (X,A) f g 1 (Y,B) f f: (X,A) (Y,B) n f : H n (X,A) H n (Y,B) X H 0 (X) = Z H n (X) = 0 (n 0) H n ( ) = 0, n Z 2. H n (X 1 X2 ) = H n (X 1 ) H n (X 2 ) 12

13 H n (X) Xn 1. n (X,A 1,...,A n 1 ) (Y,B 1,...,B n 1 ) f g 2. f 3 f 2 f 1 f 0 A 4 A3 A2 A1 A0 Imf i+1 = Kerf i, 0 i 2 (a) A 2 = 0 f 0 (b) A 0 = 0 f 1 (c) f 0 = 0, f 2 = 0 f 1 (d) f 0 f 3 A 2 = 0 3. A f B g C h D j E l m A f B g C h n p D j E q m, p l q n 4. f: X Y n f : H n (X) H n (Y) A X n H n (X,A) = 0 H n (X,A) H n (A) H n (X) 6. (x 0,{x 0 }) (X,{x 0 }) X x 0 (a) Xx 0 X X, x x 0,{x 0 } 1 X (b) X 7. X 2 X x 0 x 0 x X X x 0 (a) X x 0 X (b) Xx 0 Xx 0 13

14 X X 1 X 2 1. X = IntX 1 IntX 2 2. X 1 X 2 X X 2 X 1 k: (X 1,X 1 X 2 ) (X,X2) k : H n (X 1,X 1 X 2 ) H n (X,X 2 ) : H n (X) H n 1 (X 1 X 2 ) H n (X) j H n (X,X 2 ) k 1 H n (X 1,X 1 X 2 ) H n 1 (X 1 X 2 ) H n (X 1 X 2 ) (i 1, i 2 ) H n (X 1 ) H n (X 2 ) j 1 +j 2 Hn (X) H n 1 (X 1 X 2 ) j, k, i 1, i 2, j 1, j 2 1. λ 3n 3 λ G 3n 3 3n 2 λ G 3n 2 3n 1 G 3n 1 G 3n λ 3n G 3n+1 λ 3n+1 G 3n+2 ϕ 3n 3 ϕ 3n 2 ϕ 3n 1 ϕ 3n ϕ 3n+1 ϕ 3n+2 H 3n 3 µ 3n 3 H 3n 2 µ 3n 2 H 3n 1 µ 3n 1 H 3n µ 3n H 3n+1 µ 3n+1 H 3n+2 ϕ 3n G 3n 2 ρ H3n 2 G 3n 1 σ H3n 1 τ G3n+1 ρ H3n+1 G 3n+2 ρ = (ϕ 3n 2, λ 3n 2 )σ = µ 3n 2 +ϕ 3n 1 τ = λ 3n ϕ 1 3n µ 3n 1 2. g f G 3 H 1 λ 2 ν 1 µ 1 G 0 G 2 µ 2 λ 1 g H 2 G f 1 h ν 2 H 0 h f, f Kerλ 2 = Imλ 1 Kerµ 2 = Imµ 1 ν 2 ν 1 = 0 h f 1 g = h f 1 g 14

15 Xn H n (X) = ker(p : H n (X) H n ( )) px 8.2. X H n (X) = H n (X) (n 0), H 0 (X) = H 0 (X) Z 8.3. X n H n (X) = 0 : H n (X) H n 1 (X 1 X 2 ) : Hn (X) H n 1 (X 1 X 2 ) H n (X 1 X 2 ) (i 1, i 2 ) H n (X 1 ) H n (X 2 ) j 1 +j 2 Hn (X) H n 1 (X 1 X 2 ) 8.5. : H n (SX) H n 1 (X) H n (SX) = H n 1 (X) 8.6. H n (S m Z, n = m ) = 0, n m 1. XY x 0 X, y 0 Y X Y = X {y 0 } {x 0 } Y X Y {y 0 }Y X Y X Y Mayer-Vietoris 15

16 9 9.1 (Brower ). n f: B n B n f(x) = x x B n f: S n S n 9.2. ι H n (S n ) = Z f (ι) = kι kf degf S n r: x x/ x R n+1 {0} i: S n R n+1 {0} i = r 1 : Hn (S n ) = H n (R n+1 {0}) f: S n R n+1 {0} r f: S n S n W(f,0) f 9.3. a H n (S n ) f (a) = W(f,0) i (a) H n (R n+1 {0}) 9.4. f g: S n R n+1 {0} W(f,0) = W(g,0) 9.5 ( ). f: B n+1 R n+1 f(s n ) 0 W(f S n,0) 0 0 f(b n+1 ) 9.6. n = 0 W(f S 0,0) = 1 2 (r(f(1)) r(f( 1))) W(f S 0,0) 0 f(1) f( 1) < 0 n = 1 W(f S 1,0) 1. S 0 = {1, 1} {1} S 0, { 1} S 0 H 0 (S 0 ) = H 0 ({1}) H 0 ({ 1}) = Z Z H 0 (S 0 ) H 0 (S 0 ) Z Z 2. f: S 0 S 0 = { 1,1} degf 3. (a) g: S n R n+1 {0} W(g,0) = 0 (b) f: B n+1 R n+1 {0} W(f S n,0) = 0 16

17 10 S 1 C f, g: S 1 C {0} f g: S 1 C {0}f g(z) = f(z)g(z) f(1) = g(1) f g: S 1 C {0} f(z 2 ), z 0 f g(z) = g(z 2 ), z f, g: S 1 C {0} 1. f(1) = g(1) W(f g,0) = W(f,0)+W(g,0) 2. W(f g,0) = W(f,0)+W(g,0). 1. f(1) = g(1) = α f = f α 1, g = g α 1 f g f g W(f g,0) = W(f g,0) f(1) = g(1) = 1 f g S 1 γ S 1 S 1 f g (C {0}) (C {0}) C {0} (z,1) = (1,z) = z (z 2,1), z 0 γ(z) = (1,z 2 ), z 0 S 1 S 1 S 1 S 1 i S 1 S 1 S 1 p i : S 1 S 1 S 1 p i γ 1 S 1 H 1 (S 1 S 1 ) = H 1 (S 1 ) H 1 (S 1 ) γ (ι) = (ι,ι) (f g) (ι) = f (ι)+g (ι) = W(f,0)i (ι)+w(g,0)i (ι) = (W(f,0)+W(g,0))i (ι) 2. w S 1 C {0}c(w) f = f (1 S 1 c(1)), g = g (c(1) 1 S 1): S 1 S 1 1 S 1 c(1) 1 S 1 c(1) 1 S 1 f g f g = (f c(g(1))) (c(f(1)) g) W(f g,0) = W(f c(g(1)),0)+w(c(f(1)) g,0) = W(f,0)+W(g,0) 10.2 ( ). f(z) = 0 17

18 1. f j (1) = z 0 C {0} n f 1,, f n : S 1 C {0} f 1 f n : S 1 C {0} S 1 γn S 1 S 1 f 1 f n (C {0}) (C {0}) C {0} γ n 2(j 1)π argz n 2jπ (1 j n) z S 1 (1 j 1,z n,1 n j ) (S 1 ) j 1 S 1 (S 1 ) n j W(f 1 f n,0) = W(f 1,0)+ +W(f n,0) 2. (a) z S 1 f(z) W(f,0) = 0 (b) f, g: S 1 C {0} z f(z)/g(z)f/g W(f/g,0) = W(f,0) W(g,0) (c) n z z n z n W(z n,0) = n 3. f: B 2 B 2 g(z) = z f(z) g: B 2 C {0} z B 2 f g(z) = 0 g(s 1 ) W(g S 1,0) 0 f t: S 1 S 1 S 1 S 1 S 1 S 1 t(z 1,z 2,z 3,z 4 ) = (z1 2/z2 3,z2 2 /z2 4 ) t γ 4 : S 1 S 1 S 1 H 1 (S 1 ) H 1 (S 1 S 1 ) 6. t: S 1 S 1 S 1 t(z 1,z 2 ) = z1 2 z2 2 t γ 2: S 1 S 1 H 1 (S 1 ) H 1 (S 1 )2 a 2a 18

19 n n 1 n = 0 n+2 n+1 C n+1 C n n 1 n Cn 1 C = {C n, n } C = {C n, n} n n f n = f n 1 n f n : C n C n f = {f n}c C C = {C n, n } Z n (C) = Ker n Cn B n (C) = Im n+1 n B n (C)Z n (C) H n (C) = Z n (C)/B n (C) Cn {H n (C)}H(C) f = {f n } f n : H n (C) H n (C ) f = {f n } H(C) ) C H(C) f, g: {C n, n } {C n, n} n+1 Φ n +Φ n 1 n = g n f n Φ n : C n C n+1 Φ = {Φ n} fg Φfg fg f = g 19

20 C i C j C 0 i 0 C n j n n Cn C n 0 C = {C n, n }, C = {C n, n}, C = {C n, n}i = {i n }, j = {j n } i j H n (C ) i H n (C) j H n (C ), n Z 0 C n+1 i n+1 j n+1 C n+1 C n+1 0 n+1 n+1 n+1 0 C n i n j n Cn C n 0 n+1 n+1 n+1 0 C n 1 i n 1 j n 1 C n 1 C n 1 0 n 1 n 1 n 1 0 C n 2 i n 2 C n 2 j n 2 C n 1 0 H n (C ) = Z n (C )/B n (C ) z = [c ]c jn 1 (c ) [i 1 n 1 ( n(c))] Z n 1 (C )/B n 1 (C ) = H n 1 (C ) c, c (z ) = [i 1 n 1 ( n(c))] : H n (C ) H n 1 (C ) C i C j C 0 H n (C ) i H n (C) j H n (C ) H n 1 (C ) i K = {K n } K n K n K n = 0 K K = {K n } χ(k) = n ( 1) n rankk n C = {C n, n } χ(c) = χ(h(c)) 20

21 12 (X,A) X D = {e i } D = {e i e i A } D A AX (X,A) n 0 X n = X n A (X,A) 1. k n H k ( X n, X n 1 ) = 0 2. X A n {e n j j J n} φ j : (B n,s n 1 ) (ē n j, en j ) e n j H n ( X n, X n 1 ) = j J n Z ǫ j Z ǫ j φ j H n (B n,s n 1 ) H n ( X n, X n 1 ) H n (B n,s n 1 ) ǫ j = φ j (ι n ) C n (X,A) = H k ( X n, X n 1 ) n : C n (X,A) C n 1 (X,A) C n (X,A) = H n ( X n, X n 1 ) n H n ( X n 1 ) j H n ( X n 1, X n 2 ) = C n 1 (X,A) C(X,A) = {C n (X,A), n } H n (C(X,A)) = H n (X,A), n Z (X,A) H n (X,A) in H n ( X n,a) jn H n ( X n, X n 1 ) = C n (X,A) 1. i n Keri n = jn (B 1 n (C(X,A))) 2. j n Imj n = Z n (C(X,A)) X A n e j n 1 e k [e j : e k ] ψ j,k : H n (B n,s n 1 ) H n 1 (S n 1 ) φ j H n 1 ( X n 1 ) i Hn 1 ( X n 1, X n 2 ( l k e l )) φ 1 k H n 1 (B n 1,S n 2 ) ψ j,k (ι n ) = [e j : e k ]ι n n : C n (X,A) C n 1 (X,A) n (ǫ j ) = k J n 1 [e j : e k ]ǫ k 21

22 12.6. {H n (X)} X χ(x) χ(x) = χ(h(x)) = n ( 1)n rankh n (X) X X X n α n χ(x) = n 0 ( 1)n α n 1. G 1 r r+1 rg rankg 0 H G K 0 H K G rankg = rankh +rankk 2. C = {C n, n } χ(h(c)) = χ(c) K = {K n } χ(k) n ( 1)n rankk n 0 B n (C) Z n (C) H n (C) 0, 0 Z n (C) C n n Bn 1 (C) 0 22

23 ( ). S 2, T(n), P(n) (n = 1, 2,...) T(n), P(n) H 2 (T(n)) = Z, H 1 (T(n)) = Z 2n, H 0 (T(n)) = Z, H k (T(n)) = 0 (k 0, 1, 2) 2. H 1 (P(n)) = Z n 1 Z/2Z, H 0 (P(n)) = Z, H k (P(n)) = 0 (k 0, 1) T(n) X 1, X 2 p: B 2 B 2 / = T(n) {z 1/2 z 1}, {z z 1/2} B 2 p( B 2 )X 1 X 1 2n S 1 S 1 H 1 (X 1 ) = Z 2n, Hk (X 1 ) = 0 (k 1) X 2 X 1 X 2 = S 1 (T(n);X 1,X 2 ) 0 H 2 (T(n)) Z i 2n j Z H 1 (T(n)) 0 0 H 0 (T(n)) 0 i Z = H 1 (S 1 ) ι H1 = (X 1 X 2 ) i 1 H 1 (X 1 ) r H1 = (S 1 S 1 ) = Z 2n i P(n) X 1, X 2 X 1 Im B 2 = S 1 S 1 n (P(n);X 1,X 2 ) 0 H 2 (P(n)) Z i Z n j H 1 (P(n)) 0 0 H 0 (P(n)) 0 i Z = H 1 (S 1 ) ι H1 = (X 1 X 2 ) i 1 H 1 (X 1 ) r H1 = (S 1 S 1 ) = Z n i 2 n (2n,...,2n) 23

24 1. B 2 S 1 4n a 1 b 1 a 1 1 b 1 1 a n b n a 1 n b 1 n T(n) p: B 2 T(n) X 1 = p({z 1/2 z }), X 2 = p({z z 1/2}) (a) f = (f 1,f 2,...,f 2n 1,f 2n ): S 1 S 1 S 1 (S 1 ) 2n i. (4k 4)π/2n argz (4k 3)π/2n f 2k 1 (z) = z 4n ii. (4k 3)π/2n argz (4k 2)π/2n f 2k (z) = z 4n iii. (4k 2)π/2n argz (4k 1)π/2n f 2k 1 (z) = z 4n iv. (4k 1)π/2n argz 4kπ/2n f 2k (z) = z 4n (1 k n) g p = f g: p(s 1 ) S 1 S 1 (b) r: X 1 p(s 1 )r(z) = p(z/ z ) p(s 1 )r X 1 (c) S 1 i X 1 X 2 i 1 X1 r p(s 1 ) g S 1 S 1 p j S 1 is 1 zp(z/2) i 1 p j j (1 j 2n) (d) i 1 : H1 (X 1 X 2 ) H 1 (X 1 ) 2. B 2 S 1 2n a 1 a 1 a n a n P(n) X 1, X 2 i 1 : H1 (X 1 X 2 ) H 1 (X 1 ) H 1 (X 1 )/Imi Z n 1 Z/2Z 24

25 (Z,Q) (X,Z) [(Z,Q),(X,A)] (X,x 0 ) π(x,x 0 ) = [(S 1,1),(X,x 0 )] I = [0,1], I = {0,1} e: (I, I) (S 1,1)e(t) = cos2πt+isin2πt σ σ e e : [(I, I),(X,x 0 )] [(S 1,1),(X,x 0 )] π(x,x 0 )[(I, I),(X,x 0 )] π(x,x 0 ) [α] [β] = [α β] α β α β (S 1,1) (X,x 0 ) S 1 γ 2 S 1 S 1 α β X X X (I, I) (X,x 0 ) α(2t), 0 t 1/2 α β(t) = β(2t 1), 1/2 t 1 f: (X,x 0 ) (Y,y 0 ) f : π(x,x 0 ) π(y,y 0 ) f ([α]) = [f α] π(x,x 0 ) [α] [β] = [α β] f: (X,x 0 ) (Y,y 0 ) f : π(x,x 0 ) π(y,y 0 ) (X,x 0 ) π(x,x 0 ), f: (X,x 0 ) (Y,y 0 ) f : π(x,x 0 ) π(y,y 0 ) 14.2 X x 0, x 1 l: I X l : π(x,x 0 ) π(x,x 1 ) l ([α]) = [ l α l] π(x,x 0 ) = [(I, I),(X,x 0 )] l(t) = l(1 t) l : π(x,x 0 ) π(x,x 1 ) f g: (X,x 0 ) (Y,y 0 ) f = g : π(x,x 0 ) π(y,y 0 ) f: (X,x 0 ) (Y,y 0 ) f : π(x,x 0 ) π(y,y 0 ) X x 0 π(x,x 0 ) = 1 25

26 ιh 1 (S 1 ) = Z α: (S 1,1) (X,x 0 ) α ([ι]) H 1 (X) η: π(x,x 0 ) H 1 (X) η: π(s 1,1) H 1 (S 1 ) w: π(s 1,1) η H 1 (S 1 ) deg Z f: I S 1 f(0) = cosθ 0 +isinθ 0 θ 0 f(t) = cosθ(t)+isinθ(t), θ(0) = θ 0 ( ) θ: I S 1 2. h: I I S 1 h(0,0) = cosθ 0 +isinθ 0 θ 0 h(t,u) = cosθ(t,u)+isinθ(t,u), Θ(0,0) = θ 0 Θ: I I S ( ) θ: I R f: I S 1 e: I S 1 e(t) = cos2πt+isin2πt (0 t 1) [α] [α e] π(s 1,1) = [(I, I),(S 1,1)] f: S 1 S 1 θ f e: [0,1] S 1 degη(f) = θ(1) θ(0) 2π deg η: π(s 1,1) Z π(s 1,1) = H 1 (S 1 ) 26

27 16 m S m R n C n f f( x) = f(x), x S m f: S 1 C {0} W(f,0) 16.2 ( ). f: S 2 C f(s 2 ) 16.3 ( ). 2 f: S 2 C f(x) = f( x) x S S ( ) (3 ). S 2 3 A, B, C 1. 0 α S 1 C {0}c(α) (a) 2z 0, z 1 C {0} z 0 z 1 l: [0,1] C {0} (b) α 0 c(α) c(1) (c) 1 S 1 c(1) 1 S 1 c(1) 1 S 1 2. (a) f: S 1 C {0} W(f,0) = 0 (b) f(z) f: S 1 C {0} W(f,0) = 0 (c) n z z n z n W(z n,0) = n (d) f, g: S 1 C {0} z f(z)/g(z)f/g W(f/g,0) = W(f,0) W(g,0) 3. f, g: S 1 C {0} z S 1 2f(z), g(z) W(f,0) = W(g,0) 4. 2 f, g: S 1 C {0} 0 < g(z) < f(z) W(f +g,0) = W(f,0) 27

28 R n+1 n = {(x 0,x 1,...,x n ) x 0 + x n = 1} n n X X n X n X n S n (X) n n S n (X) = 0 n n : S n (X) S n 1 (X) S n (X) σ: n X σ j : n 1 X (0 j n) (0,x 0,...,x n 1 ), j = 0 σ j (x 0,...,x n 1 ) = (x 0,...,x j 1,0,x j,...,x n 1 ), 0 < j < n (x 0,...,x n 1,0), j = n n σ = n ( 1) j σ j S n 1 (X) j=0 S n (X) n 1 σ 1 + +n k σ k, σ j : n X, n j Z, n (n 1 σ 1 + +n k σ k ) = n 1 n σ 1 + +n k n σ k n : S n (X) S n 1 (X) S(X) = {S n (X), n } H n (S(X))Xn f: X Y X σ: n X Y S n (f)(σ) = f σ: n Y S(f) = {S n (f)}: S(X) S(Y) f = S(f) : H n (S(X)) H n (S(Y)) AX S(A)S(X) S(X,A) = S(X)/S(A) 0 S(A) i S(X) j S(X,A) 0 S( ) = 0 S(X) S(X, ) : H n (S(X,A)) H n 1 (S(A)) H n (S(A)) i H n (S(X)) j H n (S(X,A)) H n 1 (S(A)) i 28

29 (X,A) {H n (S(X,A)) n Z} : H n (S(X,A)) H n 1 (S(A)) f g: (X,A) (Y,B) f = g : H n (S(X,A)) H n (S(Y,B)) H n (S(A)) i H n (S(X)) j H n (S(X,A)) H n 1 (S(A)) i U IntA H n (S(X U,A U)) = H n (S(X,A)) P n 0 H n (S(P)) = P h: X [0,1] Y ψ j : n+1 n [0,1] (e k,0), 0 k j ψ j (e k ) = (e k 1,1), j < k n+1 n σ S n (X)(n+1) Ψ n (σ) = n ( 1) n h (σ 1) ψ j S n+1 (Y) j=0 S n (X) S n+1 (Y)Ψ n Ψ = {Ψ n } S(h 0 )S(h 1 ) n+1 Ψ n +Ψ n 1 n = S n (h 1 ) S n (h 0 ) f = g : H n (S(X)) H n (S(Y)) h: (X,A) [0,1] (Y,B) Ψ = {Ψ n } S(h 0 ) S(h 1 ): S(X,A) S(Y,B) f = g : H n (S(X,A)) H n (S(Y,B)) (S(X U) S(A))/S(A) S(X)/S(A) S(X U)/(S(X U) S(A)) = (S(X U) S(A))/S(A) 29

Part () () Γ Part ,

Part () () Γ Part , Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35