1. 1 A : l l : (1) l m (m 3) (2) m (3) n (n 3) (4) A α, β γ α β + γ = 2 m l lm n nα nα = lm. α = lm n. m lm 2β 2β = lm β = lm 2. γ l 2. 3

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1 1. 1 A : l l : (1) l m (m 3) (2) m (3) n (n 3) (4) A α, β γ α β + γ = 2 m l lm n nα nα = lm. α = lm n. m lm 2β 2β = lm β = lm 2. γ l P, Q R n = {(x 1, x 2,, x n ) ; x 1, x 2,, x n R} P = (p 1,, p n ) Q = (q 1,, q n ) P Q P Q = {(tp 1 + (1 t)q 1,, tp n + (1 t)q n ) ; 0 t 1} v 0, v 1,, v k 1

2 λ 0 v 0 + λ 1 v λ k v k = 0 (λ i R) = λ 0 = λ 1 = = λ k = 0 (6, 2, 3), (0, 5, 3), (0, 0, 7) (1, 2, 3), (1, 3, 5), (4, 3, 2) P 0, P 1,, P k R n P 0 P 1, P 0 P 2,, P 0 P k P 0, P 1,, P k R n P 0, P 1,, P k R n S(P 0,, P k ) S(P 0,, P k ) = {λ 0 OP0 +λ 1 OP1 + +λ k OPk ; λ 0 +λ 1 + +λ k = 1, λ 0,, λ k R} 3. ( ) R n k + 1 P 0,, P k ; P 0 P 1 P k = {λ 0OP0 + λ 1OP1 + + λ kopk ; λ 0 + λ λ k = 1, λ 0 0, λ 1 0,, λ k 0} P 0 P 1 P k k k ) k P 0 P 1 P k (dim P 0 P 1 P k ) R 2 P 0 = (1, 0), P 1 = (0, 1) P 0 P 1 R 3 P 0 = (1, 0, 0), P 1 = (0, 1, 0), P 2 = (0, 0, 1) P 0 P 1 P 2 s : 0 s k k P 0 P 1 P k s P 0, P 1,, P k s P 0 P 1 P 2 P 3 3 P 0 P 1 P 2 P 3 s τ k σ σ τ k σ σ σ > τ σ σ σ τ<σ τ σ Int σ = σ σ.. 0 σ σ =, Int σ = σ. 4. σ 1, σ 2,, σ k : 14 K = {σ 1,, σ k } K (1) (2) (1) σ K τ σ τ σ (σ K, τ σ = τ K) (2) σ τ K σ τ σ τ 2

3 4 4 (σ, τ K, σ τ = σ τ σ, σ τ τ). K = { P 0 P 1 P 2, P 0 P 1, P 1 P 2, P 0 P 2, P 2 P 3, P 0, P 1, P 2, P 3 } P 0, P 1, P 2, P 3, P 4 K = { P 0 P 1 P 2, P 0 P 1, P 1 P 2, P 0 P 2, P 0, P 1, P 2, P 0 P 3 P 4, P 0 P 3, P 3 P 4, P 0 P 4, P 3, P 4 } P 0 P 1 P 3 P 4 P 2 15 K (dim K) K 16 K = {σ 1, σ 2,, σ k } K K K = σ 1 σ 1 σ k 17 K dim K K dim K X : X K f : K X σ = P 0 P 1 P k : k (P 0,, P i,, P j,, P k ) (P 0,, P j,, P i,, P k ) 3

4 20 ( ) 6 (P 0, P 1, P 2 ) (P 2, P 0, P 1 ) 6 (P 0, P 1, P 2, P 3 ) (P 3, P 2, P 1, P 0 ). 21 (P i0, P i1,, P ik ) P i0, P i1,, P ik. P 0, P 1, P 2 = P 1, P 2, P 0 22 P 0, P 1, P 2,, P k, P 1, P 0, P 2,, P k P 0 P 1 P 2 P k 23 P i0, P i1,, P ik = P j0, P j1,, P jk 24 P i0, P i1,, P ik = P j0, P j1,, P jk P i0, P i1,, P ik = P j0, P j1,, P jk. 0 1 P 0, P 1 P 0, P 1 P 0 P 1 P 1, P 0 P 1 P 0 2 P 0, P 1 P 1 P 0, P 1, P 2 P 0 P 1 P 2 P 0 P 1 P 2 P 0, P 2, P 1 P 0 P 1 P 2 P 0 P 2 P 1 25 k P 0 P 1 P k P 0 P i P k 4

5 7 7 P 0 P 1 P k P 0 P i P k ( 1) i P 0 P i P k ( 1) P 0 P i P k P 0 P i P k P 0 P 1 1 P 0 P 1 0 P 0 P 1 P 2 2 P 0 P 1 P k-chain C k (K) K : m K = {σ1, 0, σi 0 0, σ1, 1, σi 1 1,, σ1 m,, σi m m } (σ j i j i ) 26 C k (K) = {n 1 σ1 k + n 2 σ2 k + + n k σi k k ; n i Z} ( ) 27 8 (n 1 σ1 k + n 2 σ2 k + + n ik σi k k ) +(n 1 σ1 k + n 2 σ2 k + + n i k σi k k ) = (n 1 + n 1) σ1 k + (n 2 + n 2) σ2 k + + (n ik + n i k ) σi k k ( 2 σ σ2 ) 2 + ( σ1 2 + σ2 ) 2 5

6 P 0 P 1 P k = k j=0 ( 1)j P 0 P 1 P j P k 8 P 0 P 1 P 2 = P 1 P 2 P 0 P 2 + P 0 P 1 9 P 0 P 1 29 m K = {σ 0 1,, σ 0 i 0, σ 1 1,, σ 1 i 1,, σ m 1,, σ m i m } : C k (K) C k 1 (K) (0 k m) (n 1 σ k 1 + n 2 σ k n ik σ k i k ) = n 1 σ k 1 + n 2 σ k n ik σ k i k k > dim K C k (K) = 0 k < 0 C k (K) = 0 C k+1 (K) = 0 C k (K) C 1 (K) C 0 (K) C 1 (K) = 0 (2 P 0 P P 1 P 2 ) ( P 0 P 1 P 2 ) : C k+1 (K) C k 1 (K) ( c C k+1 (K), (c) = 0) k cycle Z k (K) Z k (K) = {c C k (K) ; (c) = 0} (= Ker ) 32 k cocycle B k (K) B k (K) = (C k+1 (K)) = { c C k (K) ; c C k+1 (K)} 1 B k (K) Z k (K) 2 B k (K) Z k (K). Z k (K)/B k (K) 33 k H k (K) H k (K) = Z k (K)/B k (K) (B k (K) α α = 0 Z k (K) ) 10 K = { P 0 P 1, P 0, P 1 } H 0 (K), H 1 (K) P 0 P 1 Fig. 1 6

7 C 0 (K) = {n 1 P 0 + n 2 P 1 ; n 1, n 2 Z} C 1 (K) = {n P 0 P 1 ; n Z} Z 0 (K) = {c C 0 (K) ; c = 0} = {n 1 P 0 + n 2 P 1 ; (n 1 P 0 + n 2 P 1 ) = 0} = {n 1 P 0 + n 2 P 1 ; n 1, n 2 Z} B 0 (K) = { (d) ; d C 1 (K)} = { (n P 0 P 1 ) ; n P 0 P 1 C 1 (K)} = {n P 1 n P 0 ; n Z} H 0 (K) = Z 0 (K)/B 0 (K) = {n 1 P 0 + n 2 P 1 ; n 1, n 2 Z}/{n P 1 n P 0 ; n Z} P 1 P 0, 2 P 1 2 P 0, 0 P 1 = P 0 = {(n 1 + n 2 ) P 0 ; n 1, n 2 Z} = {n P 0 ; n Z} = Z P 0 = Z Z 1 (K) = {c C 1 (K) ; c = 0} = {n P 0 P 1 ; n P 0 P 1 = 0} = {n P 0 P 1 ; n P 1 n P 0 = 0} = {n P 0 P 1 ; n = 0} = {0} = 0 H 1 (K) = Z 1 (K)/B 1 (K) = {0} = 0 11 K = { P 0 P 1 P 2, P 0 P 1, P 1 P 2, P 2 P 0, P 2 P 3, P 3 P 0, P 0, P 1, P 2, P 3 } H 2 (K) P 0 P 1 P 3 Fig K H 1 (K) 12 K = { P 0 P 1 P 2, P 0 P 2 P 3, P 0 P 1, P 1 P 2, P 2 P 0, P 2 P 3, P 3 P 0, P 0, P 1, P 2, P 3 } H 1 (K) P 2 7

8 P 0 P 3 P 1 P 2 Fig. 3 1 K = { P 0 P 1, P 1 P 2, P 2 P 0, P 2 P 3, P 3 P 0, P 0, P 1, P 2, P 3 } H 1 (K) = Z Z ( ) K: β j (K) = (H j (K) rank) = (H j (K) Z ) dim K K χ(k) = ( 1) j β j (K) j=0 α j (K) = (K j ) χ(k) = dim K j=0 ( 1) j α j (K) K 2 ( )= ( ) ( ) + ( ) K, L : K 0 = (K ) L 0 = (L ) φ : K 0 L 0 K P 0 P 1 P k {φ(p 0 ), φ(p 1 ),, φ(p k )} L φ : K L φ(p 0 ),, φ(p k ) 12 K = { P 0 P 1 P 2, P 0 P 1, P 1 P 2, P 2 P 0, P 2 P 3, 8

9 L = { Q 0 Q 1, Q 1 Q 2, Q 0, Q 1, Q 2 } φ : K 0 L 0 φ(p 0 ) = Q 0, φ(p 1 ) = Q 0, φ(p 2 ) = Q 1, φ(p 3 ) = Q 2 13 K, L ψ : K 0 L 0 ψ(p 0 ) = Q 0, ψ(p 1 ) = Q 1, ψ(p 2 ) = Q 2, ψ(p 3 ) = Q 2 4 K, L, M : φ : K L : ψ : L M : = ψ φ : K M 36 K, L : φ : K L (1) φ : K 0 L 0 (2) φ 1 : L 0 K 0 37 K L φ : K L 10. K, L : φ : K L : P 0,, P k : K 38 φ(p 0 ), φ(p 1 ),, φ(p k ) φ # ( P 0, P 1,, P k ) = φ(p 0 ), φ(p 1 ),, φ(p k ) φ(p 0 ), φ(p 1 ),, φ(p k ) φ # ( P 0, P 1,, P k ) = 0 39 φ # : C k (K) C k (L) φ # (n 1 σ n l σ l ) = n 1 φ # ( σ 1 ) + + n l φ # ( σ l ) 14 K = { P 0 P 1 P 2, P 0 P 1, P 1 P 2, P 2 P 0, P 2 P 3, P 0, P 1, P 2, P 3 } L = { Q 0 Q 1, Q 1 Q 2, Q 0, Q 1, Q 2 } φ : K 0 L 0 φ(p 0 ) = Q 0, φ(p 1 ) = Q 0, φ(p 2 ) = Q 1, φ(p 3 ) = Q 2 φ # ( P 0 P P 1 P P 2 P 0 ) 2 φ # = φ # 3 φ : K L : = (1) φ # (Z k (K)) Z k (L) (2) φ # (B k (K)) B k (L) 40 Z k (K) c H k (K) = Z k (K)/B k (K) [c] 41 φ : H k (K) H k (L) 9

10 φ ([c]) = [φ # (c)] φ 13 K = { P 0 P 1, P 1 P 2, P 2 P 0, P 0, P 1, P 2 }, L = { Q 0 Q 1, Q 0, Q 1 } φ(p 0 ) = Q 0, φ(p 1 ) = Q 0, φ(p 2 ) = Q 1 φ ([ P 0 P 1 + P 1 P 2 + P 2 P 0 ]) 15 K = { P 0 P 1, P 1 P 2, P 2 P 0, P 0, P 1, P 2 } L = { Q 0 Q 1 Q 2, Q 0 Q 1, Q 1 Q 2, Q 2 Q 0, Q 0, Q 1, Q 2 } φ(p 0 ) = Q 0, φ(p 1 ) = Q 1, φ(p 2 ) = Q 2 φ ([ P 0 P 1 + P 1 P 2 + P 2 P 0 ]) 5 (1) id : K K ; = (id) : H k (K) H k (K) (2) φ : K L ; ψ : L M ; = (ψ φ) = ψ φ 10

11 (3) φ : K L ; = φ : H k (K) H k (L) : 11. X : 42 X K f : K X. 43 X H k (X) (k = 0, 1, 2, ) H k (K) 6 H k (X) 7 X, Y : X = Y = H k (X) = H k (Y ). X : H 0 (X) = Z, H k (X) = 0 (k 1). X : H 0 (X) = Z, H 1 (X) = Z, H k (X) = 0 (k 2). X : H 0 (X) = Z, H 1 (X) = 0, H 2 (X) = Z, H k (X) = 0 (k 3). X : g H 0 (X) = Z, H 1 (X) = Z 2g, H 2 (X) = Z, H k (X) = 0 (k 3) 44 X, Y : f, g : X Y ; f g F : X I Y ; (I = [0, 1] ) (1) F (x, 0) = f(x) (2) F (x, 1) = g(x) 14 X = S 1, Y = R 2 f(θ) = (cos θ, sin θ) (θ ) g(θ) = (0, 0) f g 16 X = [0, 1], Y = [0, 1] 45 f(θ) = 0, g(θ) = θ f g f g 11

12 8 f g = f = g 46 X, Y : X Y (X Y ) f : X Y ; g : Y X ; g f id (X ), f g id (Y ) 15 X = [0, 1], Y = {0} X Y 17 X = {0}, Y = D 2 X Y 9 X Y = X Y 10 X, Y : X Y = H k (X) = H k (Y ) Z : X Y H k (Z) = H k (X) H k (Y ) (k 1) H k (X) H k (Y ) {(x, y) ; x H k (X), y H k (Y )} X H 0 (X) = Z 12

13 A : l l : (1) l m (m 3) (2) m (3) n (n 3) (4) A 13

14 14 ( ) ( ) + ( ) = Brouwer D 2 : f : D 2 D 2 ; z D 2 f f(z) = z (Brouwer) f : D 2 D

15 49 K: P, Q: K P Q P 0, P 1,, P k (1) P 0 = P, P k = Q (2) P i P i+1 (i = 0, 1,, k 1) K K K P, Q P Q K C 1 (K) c = P 0 P 1 + P 1 P 2 c 15

16 20 K C 1 (K) c = P 0 P 1 + P 1 P 2 + P 2 P 5 c K H 0 (K) = Z. K K 1,, K m K H q (K) = H q (K 1 ) H q (K m ) (q 0) 16

17 15. K: K 1, K 2 : K ( K 1, K 2 K ) 21 K 1 = { P 0 P 1 P 2, P 0 P 1, P 1 P 2, P 2 P 0, P 0, P 1, P 2 } K 2 = { P 0 P 2 P 3, P 0 P 2, P 2 P 3, P 3 P 0, P 0, P 2, P 3 } K 1 K 2 K 4 K K 1, K 2 K 1 K 2 q : H q (K 1 K 2 ) H q 1 (K 1 K 2 ) [z] H q (K 1 K 2 ) c 1 C q (K 1 ) c 2 C q (K 2 ) z = c 1 + c 2 q : H q (K 1 K 2 ) H q 1 (K 1 K 2 ) q ([z]) = [ (c 1 )] z = c 1 + c 2 H q 1 (K 1 K 2 ) K = { P 0 P 1, P 1 P 2, P 0 P 2, P 2 P 3, P 0 P 3, P 0, P 1, P 2, P 3 } K 1 = { P 0 P 1, P 1 P 2, P 0 P 2, P 0, P 1, P 2 } K 2 = { P 0 P 2, P 2 P 3, P 0 P 3, P 0, P 2, P 3 } z = P 0 P 1 + P 1 P 2 + P 2 P 3 + P 3 P 0 q ([z]) 53 i : K 1 K 2 K 1 ( i(x) = x) i : K 1 K 2 K 2 ( i (x) = x) j : K 1 K 1 K 2 ( j(x) = x) j : K 2 K 1 K 2 ( j (x) = x) i, i, j, j i : H q (K 1 K 2 ) H q (K 1 ), i : H q (K 1 K 2 ) H q (K 2 ) j : H q (K 1 ) H q (K 1 K 2 ), j : H q (K 2 ) H q (K 1 K 2 ) H q (K 1 ) H q (K 2 ) = {(z, w) ; z H q (K 1 ), w H q (K 2 )} ψ q : H q (K 1 K 2 ) H q (K 1 ) H q (K 2 ) ψ q ([z]) = (i ([z], i ([z])) φ q : H q (K 1 ) H q (K 2 ) H q (K 1 K 2 ) φ q ([z 1 ], [z 2 ]) = j ([z 1 ]) + j ([z 2 ]) 18 17

18 q+1 H q (K 1 K 2 ) ψ q Hq (K 1 ) H q (K 2 ) q φ q Hq (K 1 K 2 ) H q 1 (K 1 K 2 ) ψ q 1 Hq 1 (K 1 ) H q 1 (K 2 ) q 1 φ q 1 Hq 1 (K 1 K 2 ) H 0 (K 1 K 2 ) ψ 0 H0 (K 1 ) H 0 (K 2 ) 0 φ 0 H0 (K 1 K 2 ) f : A B Im f f(a) = {f(x) B ; x A} Ker f {x A ; f(x) = 0} A 1, A 2,, A n : f 1 : A 1 A 2, f 2 : A 2 A 3, f n 1 : A n 1 A n Im f i = Ker f i+1 (1) Im q = Ker ψ q 1 (2) Im φ q = Ker q (3) Im ψ q = Ker φ q 16 5 ( ) (1) 0 f A g 0 A = 0. (2) 0 f A g B h 0 A = B ( )

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