X G P G (X) G BG [X, BG] S 2

2 2 S 2 2 S 2 = { (x 1, x 2, x 3 ) R 3 x 2 1 + x 2 2 + x 2 3 = 1 } R 3 S 2 S 2 v x S 2 x x v(x) T x S 2 T x S 2 S 2 x T x S 2 = { ξ R 3 x ξ }

R 3 T x S 2 S 2 x x T x S 2 T x S 2 x 3 v(x) x 2 x T x S 2 x 1 S 2 S 2 S 2 v x T x S 2 T S 2 = x S T 2 x S 2 T S 2 R 3 R 3 T S 2 = { (x, ξ) R 3 R 3 x S 2, x ξ } T S 2 S 2 v : S 2 T S 2

x S 2 v(x) T x S 2 T x S 2 {x} p : T S 2 S 2 v(x) T x S 2 p(v(x)) = x S 2 v : S 2 T S 2 88888888 T S 2 v p 8 S 2 = S 2 T S 2 T x S 2 T S 2 p : T S 2 S 2 S 2 T x S 2 2 x R 2 T S 2 R 2 S 2 F B E I = [0, 1] 2 D 2 = { (x, y) R 2 x 2 + y 2 1 } E = D 2 I

I D 2 A I S 1 A I S 1 = { (x, y) R 2 x 2 + y 2 = 1 } A S 1 I I S 1 A = S 1 I = I S 1 2 T 2

S 1 T 2 S 1 T 2 T 2 = {((2 + θ) φ, (2 + θ) φ, θ) 0 θ 2π, 0 φ 2π} R 3 T 2 S 1 S 1 T 2 S 1 S 1 T 2 = S 1 S 1 M M = {( (2 + t φ 2 ) φ, (2 + t φ 2 ) φ, t φ ) 2 1 2 t 1 2, 0 φ 2π} R 3 M M M I [ 1 2, 1 2 ] S1 S 1 S+ 1 S 1 M M + M M + M S+ 1 I S 1 I M

S 1 I M = I M M + = I S 1 S 1 + S 1 S 1 + B E F p : E B x B U x φ x : p 1 (U x ) = U x F p 1 (U x ) p φ x U x 88888888 U x F 1 8 1 1 E B F p φ x p : E B (B, E, F ) F E p B E B F

x U x U x = U x φ x = φ x B E F p : E B B B = α A U α α φ α : p 1 (U α ) = U α F p 1 (U α ) p φ α U α U α F 1 φ α E = B F p : E B 1 S 1 [ 1 2, 1 2 ] I 1 2 [ 1 2, 1 2 ] p : M S 1 (x, y, z) = ( (2 + t φ 2 ) φ, (2 + t φ 2 ) φ, t φ 2 ) M

p(x, y, z) = ( φ, φ) S 1 S 1 U 1 = {( θ, θ) 0 < θ < 2π} = S 1 {(1, 0)} U 2 = {( θ, θ) θ π} = {( θ, θ) π < θ < 3π} = S 1 {( 1, 0)} y U 2 U 1 S 1 O x S 1 U 1 U 2 S 1 S 1 = U 1 U 2 p 1 (U 1 ) = { ((2 + t φ 2 ) φ, (2 + t φ 2 ) φ, t φ 2 ) 1 2 t 1 2, 0 < φ < 2π} p 1 (U 2 ) = { ((2 + t φ 2 ) φ, (2 + t φ 2 ) φ, t φ 2 ) 1 2 t 1 2, π < φ < 3π}

p 1 (U 1 ) = U 1 [ 1 2, 1 2 ] f 1 : (0, 2π) [ 1 2, 1 2] p 1 (U 1 ) g 1 : (0, 2π) U 1 f 1 (φ, t) = ( (2 + t φ 2 ) φ, (2 + t φ 2 ) φ, t φ 2 g 1 (φ) = ( φ, φ) f 1 g 1 φ 1 = (g 1 1 [ 1 2, 1 2 ] ) f1 1 : p 1 (U 1 ) U 1 [ 1 2, ] 1 2 φ 1 φ 2 : p 1 (U 2 ) U 2 [ 1 2, 1 2 p : M S 1 [ 1 2, 1 2 ] p f 1, g 1 φ 1 φ 2 : p 1 (U 2 ) U 2 [ 1 2, 1 2 ] 3 S 3 1 S 1 2 S 2 n S n S n = { (x 0,..., x n ) R n+1 x 2 0 + + x 2 n = 1 } ] )

S 1 = { z C z 2 = 1 } S 2 = { (x, z) R C x 2 + z 2 = 1 } S 3 = { (z 1, z 2 ) C 2 z 1 2 + z 2 2 = 1 } p : S 3 S 2 p(z 1, z 2 ) = ( 2 z 1 2 1, 2z 1 z 2 ) p (z 1, z 2 ) S 3 p(z 1, z 2 ) S 2 p U + = S 2 {(1, 0)} U = S 2 {( 1, 0)} U + U S 2 S 2 = U + U S 2 U + U φ + : p 1 (U + ) U + S 1 φ : p 1 (U ) U S 1 (z 1, z 2 ) p 1 (U + ) ( ) φ + (z 1, z 2 ) = 2 z 1 2 1, 2z 1 z 2, z 2 z 2 ( 2 z1 2 1, 2z 1 z 2 ) U+ ( 2 z 1 2 1, 2z 1 z 2 ) (1, 0) z 2 0 φ + φ + ψ + : U + S 1 p 1 (U + ) ψ + (x, z, w) = zw 2 1 x 2 1 x, w 2

φ : p 1 (U ) U S 1 ψ : U S 1 p 1 (U ) p : S 3 S 2 S 1 p : E B S 2 p : T S 2 S 2 R 2

p : E B F B B = α A U α α φ α : p 1 (U α ) = U α F p 1 (U α ) p φ α U α U α F 1 p E = α A p 1 (U α ) p 1 (U α ) = U α F E E = U α F α A

U α F U β F U α F = p 1 (U α ) U β F = p 1 (U β ) p 1 (U α ) p 1 (U β ) φ α : p 1 (U α ) U α F φ β : p 1 (U β ) U β F (x, y) U α F (x, y ) U β F (x, y) (x, y ) E φ 1 α (x, y) = φ 1 β (x, y ) φ α φ β U α F U β F B U α F U β F (x, y) (x, y ) φ 1 α (x, y) = φ 1 β (x, y )

(U α F U β F )/ U α F U β F (x, y) (x, y ) φ β φ 1 α (x, y) = (x, y ) φ α φ β φ β φ 1 α φ β φ 1 α (U α U β ) F (U α U β ) F φ α φ β p 1 (U α U β ) (x, y) (x, y ) p 1 (U α U β ) φ β φ 1 α φ β φ 1 α (x, y) (U α U β ) F (φ β φ 1 α )(x, y) {x} F (x, y) (U α U β ) F φ β φ 1 α (x, y) {x} F 1 φ β φ 1 α (x, y) = x 1 φ β p 1 (U β ) U β F U α F p 1 (U α ) 1 1 p p U β U α φ α 1 φ β φ 1 α (x, y) = p φ 1 α (x, y) = 1 (x, y) = x

φ β φ 1 α (x, y) = ( x, Φ αβ (x)(y) ) Φ αβ (x) y F Φ αβ (x)(y) F Φ αβ (x)(y) = 2 φ β φ 1 α (x, y) x U α U β Φ αβ (x) : F F φ α φ β φ β p 1 (x) φ 1 α {x} F Φ αβ ( ) ( ) (x) = 2 φβ p 1 (x) φ 1 {x} F ix i x : F {x} F i x (y) = (x, y) x U α U β Φ αβ (x) U α U β X (X) = {f : X X } (X) Φ αβ : U α U β (F ) {Φ αβ } α

G (F ) (B, E, F ) G Φ αβ α, β G G (F ) Φ αβ Φ αβ U α U β Φ αβ G (F ) H (F ) G G H (F ) G H G U α U β U γ = (U α U β ) (U β U γ ) U α U β = U β U α (X) G (X) Φ αβ

E = B F B E = B F B F B F F F M S 1 S 1 S 1 = U 1 U 2 U 1 = {( θ, θ) 0 < θ < 2π} = S 1 {(1, 0)} U 2 = {( θ, θ) π < θ < 3π} = S 1 {( 1, 0)} φ 1 : p 1 (U 1 ) U 1 [ 1 2, 1 2 ] φ 2 : p 1 (U 2 ) U 2 [ 1 2, 1 2 ] A A B φ 1 φ 2 B A B U 1 U 2 A B Φ 12 : U 1 U 2 ([ 1 2, 1 2 ]) (U 1 U 2 ) [ 1 2, 1 2 ] φ 1 1 p 1 (U 1 U 2 ) φ 2 (U 1 U 2 ) [ 1 2, 1 2 ]

A A B φ 1 φ 2 B A B t, x A t [ 1 Φ 12 2 (x)(t) =, 1 2 ] t, x B t [ 1 2, 1 2 ] g(t) = t g ([ 1 2, 1 2 ]) Φ12 {1 [ 1 2, 1 2 ], g} { } Φ 12 : U 1 U 2 1 [ 1 2, 1 2 ], g g g(t) = g( t) = ( t) = t { } 1 [ 1 2, 1 2 ], g 2 C 2 C 2 A M S 1 I = [ 1 2, 1 2] C 2 Φ αβ : U α U β (F ) Φ αβ U α U β Φ αβ φ α φ β (F ) (F ) G

G e G µ : G G G ν : G G g G µ(e, g) = µ(g, e) = g 1 G c e G G G G 8888888888 µ 8 G c e 1 G c e : G G e g, h, k G µ(µ(g, h), k) = µ(g, µ(h, k)) G G G µ 1 G G G 1 G µ G G µ G µ g G µ(ν(g), g) = µ(g, ν(g)) = e G G G G G G µ 1 G ν c e ν 1 G c e G G G e µ(g, h) = g h g h ν(g) = g 1 g µ ν G µ

G G µ ν G G = R e = 0 µ(x, y) = x + y ν(x) = x R G = R = {x R x 0} e = 1 µ(x, y) = xy ν(x) = 1 x R R R >0 = {x R x > 0} C = {z C z 0} S 1 = {z C z = 1} M n (R) = { n n } M n (R) = R n2 M n (R) e = O 0 µ(a, B) = A + B ν(a) = A M n (R) n = 1

n (R) = {A M n (R) A 0} M n (R) e = E n µ(a, B) = AB ν(a) = A 1 n (R) n n = 1 n (R) µ ν n (R) = {A M n (R) A = 1} µ ν n (R) µ( n (R) n (R))) n (R) ν( n (R)) n (R) µ, ν n (R) n O(n) = {A M n (R) t AA = E n } O(n) n (n) = {A O(n) A = 1} = O(n) n (R) n O(n) (n) M n (C) = { n n} n (C) = {A M n (C) A 0} n (C) = {A M n (C) A = 1} U(n) = {A M n (C) t } AA = E n (n) = {A U(n) A = 1} M n (C) n (C) n (C) U(n) (n)

U(n) H R {1, i, j, k} 1 i 2 = 1 j 2 = 1 k 2 = 1 ij = ji = k jk = kj = i ki = ik = j H 0 H = H {0} H M n (H) n (H) n (H) n (H) = {A M n (H) } (n) = {A M n (H) t AA = A t } A = E n a + bi + cj + dk H a + bi + cj + dk = a bi cj dk n (H) (n) H C j j 2 = 1 R i C l 2 = 1 H 8 O

O = O {0} n (O) SL n (R) n (R) n (R) G H G H H G H G H G µ ν G H G µ(h H) H ν(h) H R >0 R S 1 C n (R) n (R) (n) O(n) n (R) n (C) n (C) U(n) n (C) (n) U(n) (n) n (H) n (R) n (C) n (H) O(n) U(n) (n) R C H G G G

G, H G H f : G H G G f f H H µ G µ H G, H G H f f g : H G µ G µ H G H f f g = 1 H, g f = 1 G f G H G = H G, H f g G H G, H f : G H f f G, H f : G H f R δ R R f : R δ R f f f : R R >0 f(x) = e x f f

f : R S 1 f(x) = e 2πix f : n (R) R r : S 1 (2) r(e iθ ) = θ θ θ θ (2) θ θ r θ θ A M n (C) A 0, A 1 A = A 0 + A 1 i r(a) = A 0 A 1 A 1 A 0 r : M n (C) M 2n (R) r r : n (C) 2n (R) r n (C) 2n (R) n (C) 2n (R) n (H) 2n (C) U(n) O(2n) (n) U(2n)

(2) = (1) (2) = a b b a a, b C, a 2 + b 2 = 1 (2) S 3 (2) (3) 2 p : E B F Φ αβ : U α U β (F ) (F ) G Φ αβ G Φ αβ (F ) (F ) (F ) (F ) Φ αβ (F ) Φ αβ (x) : F F

(U α U β ) F φ 1 α p 1 (U α U β ) φ β (U α U β ) F 2 F Φ αβ (x)(y) = 2 φ β φ 1 α (x, y) 2 φ β φ 1 α Φ αβ (x) : F F φ : X Y Z x X (φ)(x) : Y Z (φ)(x)(y) = φ(x, y) φ : X Y Z x X (φ)(x) : Y Z (φ)(x) Y y (φ)(x)(y) U y V (φ)(x)(v ) U (φ)(x)(y) = φ(x, y) φ(x, y) U φ (x, y) X Y W V φ(w V ) U (x, y) x y

(φ)(x)(v ) = φ({x} V ) φ(w V ) U (φ)(x) X Y (X, Y ) = {f : X Y } φ : X Y Z x X (φ)(x) (Y, Z) (φ) : X (Y, Z) φ Φ αβ : U α U β (F ) (U α U β ) F φ 1 α p 1 (U α U β ) φ β (U α U β ) F 2 F Φ αβ = ( 2 φ β φ 1 α ) : U α U β (F, F ) Φ αβ (F ) (F ) φ : X Y Z (φ) : X (Y, Z) (Y, Z)

Y, Z K Y U X W (K, U) = {f : X Y f(k) U } B = {W (K, U) K X : U Y : } B (Y, Z) K Y U Z W (K, U) (Y, Z) (X, Y ) (Y, Z) φ : X Y Z (φ) : X (Y, Z) x X (φ) x (φ)(x) W (φ)(v ) W x V W Y C Z U W (C, U) (φ)(x) W (C, U)

(φ)(v ) W (C, U) x V (φ)(x) W (C, U) (φ)(x)(c) U φ({x} C) U y C φ(x, y) U φ U x V y y W y φ(v y W y ) U C y C C W y W y C W y1 W yn n V = x V, y C y W yi i i=1 V yi (x, y) V W yi V yi W yi (φ)(x)(y) = φ(x, y) φ(v yi W yi ) U (φ)(v )(C) U (φ)(v ) W (C, U) F (F ) (F, F ) Φ αβ : U α U β (F )

(F ) µ : (F ) (F ) (F ) ν : (F ) (F ) F X x X V x U U U V X, Y, Z Y µ : (Y, Z) (X, Y ) (X, Z) X µ : (X) (X) (X) X X x X A X x A X U, V x U A V U V = ϕ

A V U x X (f, g) (Y, Z) (X, Y ) µ(f, g) W (f, g) W µ(w ) W (f, g) W µ 1 (W ) C X U Z W = W (C, U) µ(f, g) W µ(f, g) W (C, U) f g W (C, U) f(g(c)) U g(c) f 1 (U) y Y f 1 (U) y g(c) Y Y y g(c) y V y Y f 1 (U) W y V y W y = ϕ V y Y W y f 1 (U)

Y W y V y Y W y f 1 (U) Y V x y g(c) y V y V y : V y f 1 (U) y g(c) g(c) V y V y f 1 (U) y g(c) y g(c) C g g(c) n g(c) y 1,..., y n g(c) n V = i=1 i=1 V yi V yi V V g(c) V, f(v ) U W g W (C, V ), f W (V, U) µ(w (V, U) W (C, V )) W (C, U) W = W (V, U) W (C, V ) (f, g) W µ 1 (W (C, U)) (f, g) µ (f, g) ν : (X) (X)

X, Y B S = {W (K, U) K X U Y K Y U } B S (X, Y ) K X (X) X (X) ν : (X) (X) X (X) W (K, U) ν 1 (W (K, U)) f ν 1 (W (K, U)) f 1 (K) U K f(u) X K f(x U) f 1 (X K) X U ν 1 (W (K, U)) = W (X U, X K) ν X (X) ν ν : (X) (X) X (X) X (X) (X)

(X) X = R n (X) n (R) (R n ) n (R) ν : (X) (X) X e X µ : X X X µ (µ 1 X ) = µ (1 X µ) µ (1 X c e ) = µ (c e 1 X ) = 1 X X (X, X) (X) (R n ) n (R) n (R) (X)

f : [a, b] R [a, b] f([a, b]) R X X f(x) g(x) f : X R f, g : X R X f, g (X, R) d(f, g) = f(x) g(x) x X (X, R) d f = x X f(x) X Y = R (X, Y ) R X (Y, δ) f, g (X, Y ) d(f, g) = δ(f(x), g(x)) x X (X, Y ) d

(X, C) C(X) C C (X, Y ) X X (Y, δ) (X, Y ) {f n } n N (X, Y ) f ε > 0 N x X n > N δ(f(x), f n (x)) < ε (X, Y ) X (X, Y ) X x X δ(f(x), g(x)) X (Y, δ) (X, Y ) X K (K, Y ) Y K i K : (X, Y ) (K, Y ) K (X, Y ) ε > 0 f (K, Y ) Uε K (f) = {g (K, Y ) δ(f(x), g(x)) < ε x K} { (i K ) 1 (U K ε (f)) K X :, f (K, Y ), ε > 0 } X Y X Y (X, Y )

K X V Y W (K, V ) f W (K, V ) x K f(x) V V ε x > 0 U εx (f(x)) V U εx (f(x)) f(x) Y ε f(k) x K U ε x 2 (f(x)) K f(k) n i=1 U ε x i 2 (f(x i )) ε = ε xi i U = (i K) 1 (U K ε 2 (i K(f))) U K ε (i 2 K (f)) (K, Y ) i K (f) ε 2 g U x K f U W (K, V ) x K δ(f(x), g(x)) < ε 2 i δ(f(x), f(x i )) < ε x i 2 δ(g(x), f(x i )) < δ(g(x), f(x)) + δ(f(x), f(x i )) < ε 2 + ε x i 2 < ε x i 2 + ε x i 2 = ε xi g(x) U εxi (f(x i )) V g(k) V

U f W (K, V ) W (K, V ) U (X, Y ) U f U X K ε > 0 (i K) 1 ( U K ε (i K(f)) ) U K x K f 1 (U ε 3 (f(x))) X x K x X V x V x f 1 (U ε 3 (f(x))) V x K x K K K g n i=1 W (K i, U i ) n i=1 V x V xi U i = U ε 3 (f(x i)) K i = V xi K i K(g) U K ε (i K(f)) K = x K x K i i n i=1 K i δ(f(x), g(x)) δ(f(x), f(x i )) + δ(f(x i ), g(x)) < ε K 3 + δ(f(x i), g(x))

V xi f 1 (U ε 3 (f(x i))) V xi U 2ε 3 (f(x i)) δ(f(x), g(x)) < ε 3 + 2ε 3 = ε i K( n i=1w (K i, U i )) U K ε (i K(f)) U X, Y : (X, Y ) X Y (f, x) = f(x) X ev (X, Y ) [n] = {1, 2,..., n} e : ([n], X) X n e(f) = (f(1), f(2),..., f(n)) M n (R) m n M m,n (R) R mn R mn A M m,n (R) A : R n R m M m,n (R) (R n, R m )

(R n, R m ) (F ) (F ) (F ) F G X G X µ : G X X x X µ(e, x) = x g, h G x X X c e 1 X = G X X µ(g, µ(h, x)) = µ(gh, x) µ G G X 1 G µ G X µ G 1 X G X µ G : G G G G µ X µ µ(g, x) = g x X G X

G G µ : G G G µ G G µ G µ G G G R n G G G H G µ µ H G : H G G H G G µ c : G G G (g, h) G G µ c (g, h) = ghg 1 G G G = n (R) X = R n G X µ(a, v) = Av n (R) R n X Σ n n µ : X n Σ n X n

(x 1,, x n ) X n σ Σ n µ((x 1,, x n ), σ) = (x σ(1),, x σ(n) ) Σ n X n Σ n G X µ X : G X X µ X : X G X µ X (x, g) = (g 1, x) G = O(n), X = S n 1 O(n) n (R) S n 1 R n n (R) R n O(n) S n 1 A O(n), v S n 1 Av S n 1 v S n 1 v 2 = 1 v, v = 1 t vv = 1 v n 1 Av S n 1 t (Av)(Av) = 1 t (Av)(Av) = t v t AAv O(n) t AA = E n

G X µ : G X X H G A X µ(h A) A µ H A H A n = 2 O(2) S 1 S 1 O(2) A O(2) A = ±1 : O(2) {±1} O(2)/ = {±1} = (2) T O(2) (2) O(2) = (2) T (2) A O(2) A (2) B (2) A = T B (2) = θ θ θ θ 0 θ < 2π T = 1 0 0 1 y O(2) S 1 y (2) xy (n) (n) R n

O(2) l R 2 T l l y T l (v) l v 0 x T l T l O(2) T l T (2) 0 θ < 2π R θ = θ θ θ θ n 3 D 2n = R 2π n T O(2) D 2n O(2) R 2 D 2n y n D 2n 2n D 2n O(2) µ : G X X (µ) : G (X, X)

y R 2π n 2π n 0 x G X µ : G X X G X µ (µ) : G (X, X) (X) (µ) (µ) : G (X) g G, x X gx = µ(g, x) g G (µ)(g) : X X (µ)(g 1 )

x X (µ)(g 1 ) (µ)(g)(x) = (µ)(g 1 )(µ(g, x)) = (µ)(g 1 )(gx) = µ(g 1, gx) = g 1 (gx) = (g 1 g)x = ex = x (µ)(g 1 ) (µ)(g) = 1 X (µ)(g) (µ)(g 1 ) = 1 X (µ)(g) G X µ : G X X (µ) : G (X) (µ) (µ) g, h G (µ)(gh) = (µ)(g) (µ)(h) x X (µ)(gh)(x) = µ(gh, x) = (gh)x (µ)(g) (µ)(h)(x) = (µ)(g)(µ(h, x)) = (µ)(g)(hx) = µ(g, hx) = g(hx) = (gh)x

(µ)(gh) = (µ)(g) (µ)(h) G X X G (X) G X φ : X (Y, Z) 1 (φ) : X Y Z 1 (φ)(x, y) = φ(x)(y) 1 : (X Y, Z) (X, (Y, Z)) 1 (X Y, Z) (X, (Y, Z)) : (X Y, Z) (X, (Y, Z)) Y φ : X (Y, Z) 1 (φ) : X Y Z

U Z (x 0, y 0 ) ( 1 (φ)) 1 (U) = {(x, y) X Y φ(x)(y) U} ( 1 (φ)) 1 (U) φ(x0 ) : Y Z φ(x 0 ) 1 (U) Y φ(x 0 ) 1 (U) y 0 Y Y y 0 V V (Y φ(x 0 ) 1 (U)) = φ(x 0 )(V ) U Y V φ 1 (W (V, U)) V (x 0, y 0 ) 1 (φ) φ 1 (W (V, U)) V ( 1 (φ)) 1 (U) G X φ : G (X) 1 (φ) : G X X G X X 1 (φ) 1 (φ) X 1 X (X) X 1 (X) : (X) (X) (X) X

X G X G (X) F G G (F ) F G F G (F ) G F G (F ) (B, E, F ) G {Φ αβ } α, β G (F ) Φ αβ Φ αβ U α U β Φ αβ S 3 p S 2 S 1 U + = S 2 {(1, 0)} U = S 2 {( 1, 0)}

φ + : p 1 (U + ) U + S 1 φ : p 1 (U ) U S 1 φ 1 + : U + S 1 p 1 (U + ) φ + (z 1, z 2 ) = φ (z 1, z 2 ) = φ 1 + (x, z, w) = ( ) 2 z 1 2 z 2 1, 2z 1 z 2, z 2 ( ) 2 z 1 2 z 1 1, 2z 1 z 2, z 1 zw 1 x, w 2 2 1 x 2 φ φ 1 + : (U + U ) S 1 (U + U ) S 1 φ φ 1 + (x, z, w) = φ zw 2 = = = 2 zw 2 4( 1 x 1 x 2 1 x, w 2 2 ) 1, 2 zw 2 ( z 2 1 x 1, z, z ( ) x, z, z z w ) z w 1 x 2 1 x w, 2 zw 2 1 x 2 zw 2 1 x 2 Φ + : U + U (S 1 ) Φ + (x, z)(w) = z z w z z S1 Φ + (x, z) = z z Φ + : U + U S 1 (µ) S 1 (S 1 ) Φ + Φ + U + U

(µ) : S 1 (S 1 ) S 1 S 1 S 1 (x, z) U + U z z S1 H H G G ξ = (B, E, F ) G Φ αβ : U α U β G G H ξ H G X X/G X/G = {Gx x X} Gx = {gx g G} X x G p : X X/G p(x) = Gx X/G O O = {U X/G p 1 (U) X } U X/G p 1 (U) X X/G X G

G G\X X/G X/G H G G H G G H G H X X X/ X/ = {[x] x X} p : X X/ p(x) = [x] x X [x] x X/ O O = {U X/ p 1 (U) X } X/ X O G X X G x, x X x G x g G x = gx G G X X/ G = X/G x X x [x] [x] = {x X x G x } = {gx g G} = Gx

X/ G = {[x] x X} = {Gx x X} = X/G X/G X G X/G = X G X/G X = S 1 = {z C z = 1} G = C 2 = {e, g} 2 e G G X e z = z g z = z X/G g z S 1 z S 1 + 1 1 S 1 C 2 S 1 S 1 S 1 = S 1 + S 1 {1} { 1} g S+ 1 S 1 1 1 G S+ 1 S 1 1 1 S 1 /G = S 1

S 1 /G S 1 f : S 1 /G S 1 f([z]) = z 2 f f f f 1 z, z X z G z z = z z = z f f U S 1 f 1 (U) S 1 /G S 1 /G p 1 (f 1 (U)) p 1 (f 1 (U)) = (f p) 1 (U) f p : S 1 S 1 (f p)(z) = z 2 p 1 (f 1 (U)) f f : X Y f U Y U f 1 (U)

f : X Y X f x f x f(x) = f(x ) Y = X/ f π π X Y f g Z g f f [z], [z ] S 1 /G f([z]) = f([z ]) z 2 = (z ) 2 z = ±z z G z f f w S 1 0 θ < 2π w = e iθ z = e iθ 2 z S 1 f([z]) = w f f f 1 f X Y f : X Y f f S 1 /G X X X X/

S 1 /G = S 1 X = S 1 G = C p = {e, g, g 2,..., g p 1 } p G X X/G = S 1 g k x = e 2πik p x 2 C 2 S n = { (x 0,..., x n ) R n+1 x 2 0 + + x 2 n = 1 } g(x 0,, x n ) = ( x 0,, x n ) R n n S 3 S 2n+1 = { (z 0,, z n ) C n+1 z 0 2 + + z n 2 = 1 } S 2n+1 C n+1 g C p C p S 2n+1 (z 0,, z n ) S 2n+1 g(z 0,, z n ) = L 2n+1 p ) (e 2πi p z0,, e 2πi p zn S 1 S 2n+1 ω S 1 (z 0,, z n ) S 2n+1 ω(z 0,, z n ) = (ωz 0,, ωz n ) C n n C 1 S 2 S 3 /S 1 = S 2 p : S 3 S 2

R n C n R C R n+1 {0} C n+1 {0} R (R n+1 {0}) R n+1 {0} C (C n+1 {0}) C n+1 {0} ω (x 0,, x n ) = (ωx 0,, ωx n ) (R n+1 {0})/R = R n (C n+1 {0})/C = C n R 3 = (3) O(n) = {A M n (R) t } AA = 1 n S n 1 = {(x 1,, x n ) R n x 2 1 + + x 2 n = 1 } O(n) S n 1 O(n) S n 1 S n 1 n = 2O(2) S 1 O(2) S 1 S 1

S 1 /O(2) O(2) (2) = {A O(2) A = 1} = θ θ θ θ 0 θ < 2π R θ = θ θ θ θ v, w S 1 v w θ R θ v = w R θ w v θ θ S 1 2 O(2) S 1 S 1 /O(2) = { } O(2) (2) S 1 /(2) = { } S n 1 /SO(n) = { }

G X x, y X x = gy g G G G G X X/G G H G G/H µ : G G/H G/H µ(g, g H) = gg H µ µ µ G G µ G 1 G p G G/H µ p G/H p µ 1 G p µ p 1 G 1 G p Q R Q N N p : Q Q/ N 1 Q p : Q Q Q (Q/ N ) 1 G p

f : X Y f 1 (f(v )) = V V X x V X U x U U = f 1 (f(u)) U U V g : X Y f g : X X Y Y X Y f : X Y g : Z W f g : X Z Y W X f : Y Z 1 X f : X Y X Z G G/H G G G/H G G/H R/Q G H G/H X G X G/H

G X µ : G X X x 0 X H = {g G gx 0 = x 0 } H G φ(gh) = gx 0 φ : G/H X G X µ X 1 G φ φ G G/H µ G/H G/H µ G/H G X X G G/H H φ : G X φ(g) = gx 0 φ H = φ 1 ({x 0 }) X H G h, h H (hh )x 0 = h(h x 0 ) = hx 0 = x 0 hh H h H hx 0 = x 0 h 1 (hx 0 ) = h 1 x 0 x 0 = h 1 x 0 h 1 H H G H φ φ φ G x X gx 0 = x g G g φ(gh) = gx 0 = x

φ φ(gh) = φ(g H) gx 0 = g x 0 x 0 = g 1 g x 0 g 1 g H gh = g H φ φ = φ p G p G/H φ φ p φ 1 φ X H = {g G gx 0 = x 0 } G x 0 x0 (G) O(n) O(n) M n (R) = R n2 O(n) R n2 A O(n) A = (a ij ) t A = (b ij ) A t A = E n n 1 i = k a ij b jk = 0 i k j=1

b ij = a ji n 1 i = k a ij a kj = 0 i k j=1 i n a 2 ij = 1 j=1 n a 2 ij = n i,j=1 O(n) R n2 n f(a) = A t A f : M n (R) M n (R) O(n) = f 1 ({E n }) M n (R) = R n2 O(n) {E n } O(n) S n 1 µ : O(n) S n 1 S n 1 (A, v) µ(a, v) = Av S n 1 R n O(n) v S n 1 v R n {a 1, a 2,, a n 1, a n = v} A = t a 1 t a n

a 1,, a n 1 n t a 1 A t A = (a 1,, a n ) = t a n t a 1 a t 1 a 1 a 2 t a 1 a n t a 2 a t 1 a 2 a n t a n a t 1 a n a 2 t a n a n {a 1,, a n } t a i a j a i a j t 1 i = j a i a j = 0 i j A t A = 1 n A O(n) A t a 1 v t a 1 a n Av = = t a n v t a n a n 0 = 0 1 0 e n = 0 1 v S n 1 A O(n) Av = e n u S n 1 Bu = e n B O(n) A 1 Bu = A 1 e n = v A 1 B O(n) O(n) S n 1 v S n 1 v (O(n)) = {A O(n) Av = v}

S n 1 = O(n)/v (O(n)) v (O(n)) v S n 1 v = e n A = (a ij ) en (O(n)) Ae n = e n a 11 a 1n a n 11 a n 1n a n1 a nn 0 0 1 = 0 0 1 a 1n a n 1n a nn = 0 0 1 A = a 11 a 1n 1 0 a n 11 a n 1n 1 0 a n1 a nn 1 1 A t A = E n a 11 a 1n 1 0 a n 11 a n 1n 1 0 a n1 a nn 1 1 a 11 a n 11 a n1 a 1n 1 a n 1n 1 a nn 1 0 0 1 = 1 0 0 0 0 0 0 1 a 2 n1 + + a 2 nn 1 + 1 = 1 a n1 = = a nn 1 = 0

0 A = A 0 0 0 1 a 11 a 1n A = At A = E n A t A = a n 11 a n 1n 1 E n 1 A en (O(n)) A O(n 1) 0 A = A 0 0 0 1 A O(n 1) 0 A en (O(n)) 0 0 0 1 A 0 A 0 0 0 1 O(n 1) O(n) en (O(n)) = O(n 1) S n 1 = O(n)/O(n 1) e n S n 1 en (O(n)) G X x, y X x (G) y (G) G

U(n) = {A M n (C) A t A = E n } S 2n 1 = { (z 1,, z n ) C n z 1 2 + + z n 2 = 1 } U(n) S 2n 1 U(n) U(n) S 2n 1 U(n)/U(n 1) = S 2n 1 G H G G/H H O(n) O(n)/O(n 1) = S n 1 O(n)/O(n 1) = S n 1 p : O(n) S n 1 p(a) = Ae n e n = t (0,, 0, 1) p(a) = Ae n p : O(n) S n 1 O(n 1) S n 1 {U α } s α : U α p 1 (U α ) p s α = 1 Uα α φ α (A) = (p(a), s α (p(a)) 1 A) φ α : p 1 (U α ) U α O(n 1)

φ α ψ α : U α O(n 1) p 1 (U α ) ψ α (x, A) = s α (x)a φ α φ α : p 1 (U α ) U α O(n 1) φ α (A) = (p(a), s α (p(a)) 1 A) s α U α O(n) s α (p(a)) 1 A O(n 1) s α (p(a)) 1 A en (O(n)) s α (p(a))e n Ae n p p s α = 1 Uα s α (p(a))e n = p(s α (p(a))) = p(a) = Ae n s α (p(a)) 1 Ae n = e n s α (p(a)) 1 A O(n 1) = en (O(n)) ψ α p(ψ α (x, A)) = s α (x)ae n = s α (x)e n = p(s α (x)) = x ψ α (x, A) p 1 (U α ) (x, A) U α O(n 1) φ α ψ α (x, A) = φ α (s α (x)a) = (p(s α (x)a), s α (p(s α (x)a) 1 s α (x)a)) A O(n 1) = en (O(n)) p(s α (x)a) = s α (x)ae n = s α (x)e n = p(s α (x)) = x φ α ψ α (x, A) = (x, s 1 α s α (x)a) = (x, A)

A p 1 (U α ) ψ α φ α (A) = ψ α (p(a), s α (p(a)) 1 A) = s α (p(a))s α (p(a)) 1 A = A φ α S n 1 U + = S n 1 {e n } U = S n 1 { e n } s + : U + p 1 (U + ) x = (x 1,, x n ) U + s + (x) = 1 x 1x 1 1 x n x 1x 2 1 x n x 1x n 1 1 x n x 1 x2x1 1 x n 1 x2x2 1 x n x 2 xn 1x1 1 x n 1 xn 1xn 1 1 x n x n 1 x 1 x 2 x n 1 x n s + (x) O(n) p s + (x) = s + (x)e n = x s + s s + 1 C = 1 0 0 1 1 x U s (x) = C 1 s + (Cx) s + (x) O(n) s +

s α : U α p 1 (U α ) p s α = 1 Uα G H p : G G/H s : G/H G p s = 1 G/H φ(g) = (p(g), s(p(g)) 1 g) φ : G G/H H G s φ p : E B s : B E p s = 1 B p p : G G/H p s ± = 1 U± s ± : U ± p 1 (U ± ) p : E B p x B U x s x : U x p 1 (U x ) p s x = 1 Ux

G H p : G G/H H O(n) O(n)/O(n 1) G H G G/H G G H p : G G/H p : G G/H p : G G/H U α U β G/H U α U β ϕ s α, s β p 1 (U α U β ) s α : U α p 1 (U α ) s β : U β p 1 (U β ) φ α : p 1 (U α ) U α H φ β : p 1 (U β ) U β H x U α U β h H y φ 1 α (x, h) = s α (x)h φ β (y) = (p(y), s β (p(y)) 1 y) φ β φ 1 α (x, h) = φ β (s α (x)h) = (p(s α (x)h), s β (p(s α (x)h)) 1 s α (x)h) = (x, s β (x) 1 s α (x)h)

Φ αβ : U α U β (H) Φ αβ (x)(h) = s β (x) 1 s α (x)h (µ H ) : H (H) H Φ αβ : U α U β H Φ αβ (x) = s β (x) 1 s α (x) H Φ αβ (µ H ) U α U β (H) Φ αβ H H G H p : G G/H G G/H H H H H H G p : E B G G G G G H p : G G/H H O(n) O(n)/O(n 1) O(n)/O(n 1) = S n 1 O(n) S n 1 O(n 1)

p : S 3 S 2 S 1 G = S 1 = {z C z = 1} H = C 2 = {1, 1} G S 1 /C 2 = S 1 C 2 p : S 1 S 1 /C 2 = S 1 p S 1 C 2 C 2 S 1

G G p : E B G F B {U α } φ α : p 1 (U α ) = U α F Φ αβ : U α U β G U α F Φ αβ : U α U β G G F p : E B B {U α } Φ αβ : U α U β G G F G (F ) G G G G G G (G)

G p : P B B {U α } Φ αβ : U α U β G G E B G P B G G F F G G G F G (F ) {B G } {B F G } G G p : P B G G P P G P {φ α : p 1 (U α ) U α G} α A x P, g G x p 1 (U α ) φ α (x) = (p(x), φ α (x)) x g = φ 1 α (φ α (x)g) = φ 1 α (p(x), φ α (x)g) g x p 1 (U α ) G p 1 (U α ) p 1 (U α ) G φα id (U α G) G = U α (G G) id µ U α G φ 1 α p 1 (U α ) P = α A p 1 (U α ) P G P

x p 1 (U α ) p 1 (U β ) φ α φ β x g x p 1 (U α ) p 1 (U β ), g G φ α : p 1 (U α ) U α G φ β : p 1 (U β ) U β G φ α (x) = (p(x), φ α (x)) φ β (x) = (p(x), φ β (x)) p 1 (U α ) x g = φ 1 α (p(x), φ α (x)g) p 1 (U β ) x g = φ 1 β (p(x), φ β(x)g) φ β φ 1 α (p(x), φ α (x)g) = (p(x), φ β (x)g) Φ αβ : U α U β G (y, h) (U α U β ) G φ β φ 1 α (y, h) = (y, Φ αβ (y)h), φ β φ 1 α (p(x), φ α (x)g) = (p(x), Φ αβ (p(x))φ α (x)g) φ β (x) = Φ αβ (p(x))φ α (x) φ β φ 1 α (p(x), φ α (x)) = φ β φ 1 φ α (x) = φ β (x) = (p(x), φ β (x)) φ β φ 1 α (p(x), φ α (x)) = (p(x), Φ αβ (p(x))φ α (x)) α X Y f : X Y X X = α A U α f U α f Uα f

G φ α : p 1 (U α ) U α G G G U α G (x, h) g = (x, hg) P G P /G P /G p : P B G p p : P /G = B π : P P /G p : P B P = P π P /G p B p p : P /G B p([x]) = p(x) [x] x P P /G p p p p p p [x] = [y] x = yg g G p(y) U α φ α = p φ α : p 1 (U α ) U α G

yg yg = φ 1 α (p(y), φ α (y)g) p 1 (U α ) p φ 1 α U α U α G 1 1 p(x) = p(yg) = p φ 1 α (p(y), φ α (y)g) = 1 (p(y), φ α (y)g) = p(y) p p p p p p x, y P p([x]) = p([y]) p(x) = p(y) b = p(x) = p(y) b g = φ α (y) 1 φ α (x) φ α = p φ α : p 1 (U α ) U α G φ α (yg) = (p(yg), φ α (yg)) = (p(y), φ α (y)g) = (b, φ α (x)) = φ α (x) φ α x = yg [x] = [y] p p p : E B p : S 3 S 2 S 1 S 1 S 3 S 2 U + = S 2 {(1, 0)} U = S 2 {( 1, 0)} p

φ + : p 1 (U + ) U + S 1 φ : p 1 (U ) U S 1 φ + (z 1, z 2 ) = φ (z 1, z 2 ) = ( ) 2 z 1 2 1, 2z 1 z 2, z2 z 2 ( ) 2 z 1 2 1, 2z 1 z 2, z 1 z 1 S 1 S 3 x p 1 (U ± ), w S 1 x w = φ 1 ± (p(x), φ ± (x)w) φ ± φ ± (x) = (p(x), φ ± (x)) x w x p 1 (U + ) x = (z 1, z 2 ) C 2 φ 1 + (x, z, w) = φ + (z 1, z 2 ) = z 2 z 2 zw 2 (z 1, z 2 ) p 1 (U + ), w S 1 1 x 2 1 x, w 2 (z 1, z 2 ) w = φ 1 + (p(z 1, z 2 ), φ + (z 1 )w) ( ) = φ 1 + 2 z 1 2 1, 2z 1 z 2, z 2 z 2 w = 2z z 1z 2 2 z 2 w z 2 1, 1 (2 z 2 1 2 1) z 2 w (2 z1 2 1) 2 2 ( ) z 1 z 2 w = 1 z1, z 2 2 z 2 w 1 z 1 2 = (z 1 w, z 2 w) S 1 p 1 (U + ) S 1 p 1 (U ) S 3 /S 1 = C 1

S 3 /S 1 = S 2 C 1 = S 2 G p : P B G P G + G F = G F G P B G F G P F G F G G P F G P F µ : G (P F ) P F µ(g, x, y) = (xg 1, gy) P G F (x, y) P F P G F [x, y] µ(g, x, y) = (xg, gy) p : P B G F G p : P G F B p([x, y]) = p(x) F G {B G } {B F G } (P B) (P G F B)

G p : P B {φ α : p 1 (U α ) U α G} p : P G F B {ψ α : p 1 (U α ) U α F } p 1 (U α ) = {[x, y] P G F p([x, y]) U α } = {[x, y] P G F p(x) U α } = p 1 (U α ) G F p 1 (U α ) = p 1 (U α ) G F φα Gid (U α G) G F [(x, g), y] (x, gy) 1 Uα µ : (U α G) G F U α F µ G F (x, y) [(x, e), y] ψ α : p 1 (U α ) U α F p 1 (U α ) = p 1 (U α ) G F φ α G id (U α G) G F µ U α F φ α 1 Uα µ ψ α ψ α p 1 (U α ) p ψ α U α U α F pr 1 p : P G F B

(x, y) (U α U β ) F ψ β ψ 1 α (x, y) = ψ β (φ 1 α G id) ( µ) 1 (x, y) = ψ β (φ 1 α G id)(x, e, y) = ψ β (φ 1 α (x, e), y) = ( µ) (φ β G id)(φ 1 α (x, e), y) = ( µ)(φ β φ 1 α (x, e), y) = ( µ)(x, Φ αβ (x)e, y) = (x, Φ αβ (x)y) Φ αβ : U α U β G p : P B p : P G F B Φ αβ G (P B) (P G F B) P B F C 2 p : S 1 S 1 /C 2 = S 1 S 1 = { z C z 2 = 1 } p(z) = z 2 C 2 = {1, 1} S 1 C 2 [ 1 2, 1 2 ] 1 t = t ( 1) t = t S 1 C2 [ 1 2, 1 2 ] = M S 1 [ 1 2, 1 2 ] C 2 ( 1)(z, t) = ( z, t)

S 1 [ 1 2, 1 2 ] C 2 S 1 C2 [ 1 2, 1 2 ] S 1 C2 [ 1 2, 1 2 ] S1 M S 1 S 1 C2 [ 1 2, 1 2 ]

= = G G/H H G p : P B G s : B P P = B G S 1

I M S 1

{B G} {B F G } {B G} B G B B B ξ = (F E p B) ξ = (F E p B ) ξ ξ B B,

E E, F F ξ = (p : E B) ξ = (p : E B ) ξ ξ f : E E f : B B f E E p p f B B ( f, f) : (E, B) (E, B ) f = ( f, f) : ξ ξ x B f x p 1 (x) f(x) p 1 (f(x)) p 1 (x) E B f p 1 (x) p 1 (f(x)) f E f B

p : E B p : E B {φ λ : p 1 (U λ ) U λ F } {ψ µ : p 1 (V µ ) V µ F } Φ λλ : U λ U λ (F ) Ψ µµ : V µ V µ (F ) ( f, f) : (E, B) (E, B ) ( f, f) f f : U λ U λ f 1 (V µ V µ ) V µ V µ Φ λλ U λ U λ f 1 (V µ V µ ) (F ) f Ψ µµ V µ V µ (F ) (F ) (F ) f f F : F F

x U λ U λ f 1 (V µ V µ ) Φ λλ Φ λλ (x) : F F Ψ µµ (f(x)) : F F f F Φ λλ (x) F f F Ψ µµ (f(x)) f F F F f F Ψ µµ (f(x)) = ( f F ) Φ λλ (x) ( f F ) 1 f : X Y c f : (X) (Y ) c f (φ) = f φ f 1 c f φ f 1 φ f f : E E f F : F F c f F : (F ) (F )

ξ = (p : E B) ξ = (p : E B ) F ξ ξ ξ ξ ( f, f) : (E, B) (E, B ) f x B f p 1 (x) : p 1 (x) p 1 (f(x)) ξ = (p : E B) ξ = (p : E B ) F G G F µ : G F F ξ ξ ξ ξ B {U λ } λ Λ B {V µ } µ M U λ V µ φ λ : p 1 (U λ ) U λ F ψ µ : p 1 (Vµ ) V µ F (U λ f 1 (V µ )) F φ 1 λ p 1 (U λ f 1 (V µ )) (F ) f p 1 (f(uλ ) V µ ) ψµ (f(u λ ) V µ ) F 2 F L f U λ V µ : U λ f 1 (V µ ) (F ) G (µ) G (F ) Lf U λ Vµ U λ f 1 (V µ ) L f U λ Vµ

2 ψ µ f φ 1 λ ( 2 ψ µ f φ 1 λ ) : U λ f 1 (V µ ) (F, F ) (F ) G ξ = (p : E B) ξ = (p : E B ) G f = ( f, f) : (E, B) (E, B ) E G f 1 G E G f E E G G G X Y G f : X Y G G G X 1 G f G Y X f Y G ξ = (p : E B) ξ = (p : E B) F B f = ( f, f) : (E, B) (E, B)

f = 1 B ξ ξ f ξ = ξ ξ ξ G G F f G = (F ) = ξ ξ = ξ ξ = η η = ξ ξ = ζ ζ = η ξ = η ξ = (p : E B) E = E B p = B p ξ ξ = (p : E B) ζ = (p : E B) η = (p = E B) f E E E E p p p p = = B B B B ξ ζ ζ η g f E E g B p = B p ξ η

f E E B p = B p ξ = (p : E B) η = (p : E B) E g E B p = B p g = f 1 f f 1 f 1 B = α A U α B U α ξ η φ α : p 1 (U α ) U α F ψ α : p 1 (Uα ) U α F U α E = p 1 (Uα ) α A p 1 (U α ) f 1 U α F φ 1 α p 1 f (U α ) p 1 (Uα ) ψα U α F ψ α f φ 1 α (x, y) = (x, Φ(x)(y)) Φ : U α (F ) x U α Ψ(x) = Φ(x) 1 Φ(x) 1 Φ(x) (F ) Ψ Ψ : U α Φ ν (F ) (F ) ν G Ψ g α : U α F U α F

g α (x, y) = (x, Ψ(x)(y)) ψ α f φ 1 α (x, y) = (x, Φ(x)(y)) p 1 (Uα ) ψ α U α F f 1 p 1 (U α ) φ α g α U α F f 1 f 1 ( f, 1 B ) : (E, B) (E, B) ( f 1, 1 B ) ( f, 1 B ) : (E, B) (E, B) f : E E M 0 M 2 2π M 2 M 0 = { (2 φ, 2 φ, t) 1 2 t 1 2, 0 φ 2π} M 2 = { ((2 + t φ) φ, (2 + t φ) φ, t φ) 1 2 t 1 2, 0 φ 2π} p 0 : M 0 S 1 p 2 : M 2 S 1 I [ 1 2, 1 2 ] M 0 M 2

M 0 M 2 AB CD M 2 E 1 E 2 M 0 A B C D E 1 E 2 A D A D B C B C M 2 M 0 M 2 ABDC A B C D f 1 : E 1 E 1 f 2 : E 2 E 2 AB CD f 1 f 2 M 2 M 0 p 2 p 0 = S 1 S 1 M n nπ p n : M n S 1 M 2n = M 2n+1 =

M 2 M 0 f 1 f 2 M 2 M 0 M 2 M 0 f 1 f 2 M 2 3 R 3 R 3 E p B E p B F G G F {U α } α A B U α E E { Φ αβ : U α U β G } { Ψ αβ : U α U β G } E E E E α A α, β A x U α U β λ α : U α G λ β (x) 1 Ψ αβ (x)λ α (x) = Φ αβ (x) E, E φ α : p 1 (U α ) U α F ψ α : p 1 (Uα ) U α F

f : E E x U α U β, y F 1 ψ β fφ α (x, y) = (x, λ αβ (x)(y)) λ αβ = (µ) λ αβ λ αβ : U α U β G λ α = λ αα (U α U β ) F (U α U β ) F φ α ψ β p 1 f (U α U β ) p 1 (Uα U β ) φ β ψ α (U α U β ) F (U α U β ) F p 1 (U α U β ) ψ α ψ β f φ 1 α (x, y) = ψ β ψα 1 ψ α f φ 1 α (x, y) = ψ β ψα 1 (x, λ α (x)(y)) = (x, Ψ αβ (x)λ α (x)(y)) ψ β f φ 1 α (x, y) = ψ β f φ 1 β φ β φ 1 α (x, y) = ψ β f φ 1 β (x, Φαβ (x)(y)) = (x, λ β (x)φ αβ (x)(y)) λ β (x)φ αβ (x) = Ψ αβ (x)λ α (x) Φ αβ (x) = λ β (x) 1 Ψ αβ (x)λ α (x) λ α : U α G f α : p 1 (U α ) p 1 (Uα )

p 1 (U α ) φα U α F Λα U α F ψ 1 α p 1 (Uα ) Λ α (x, y) = (x, λ α (x)(y)) x p 1 (U α U β ) λ β (x)φ αβ (x) = Ψ αβ (x)λ α (x) f : E E f : X Y X {U α } α A f Uα f Y V f 1 (V ) = f 1 (V ) U λ = (f Uα ) 1 (V ) α A α A U α X (f Uα ) 1 (V ) U α (f Uα ) 1 (V ) X f 1 (V ) X ξ = (p : E B) f : X B f (E) = {(x, e) X E f(x) = p(e)} f (p) : f (E) X p (f) : f (E) E

f (p)(x, e) = x p (f)(x, e) = e f (E) f (p) X p (f) f E B p f (E) f (p) : f (E) X f (ξ) ξ f f (p) : f (E) X p : E B F G {U α } α A B U α p : E B φ α : p 1 (U α ) U α F φ α (e) = (p(e), φ α (e)) f (E) X V α = f 1 (U α ) {V α } α A X ψ α : f (p) 1 (V α ) V α F ψ α (x, e) = (x, φ α (e)) f (p) 1 (V α ) V α p 1 (U α ) ψ α f (p) 1 (V α ) V α p 1 (U α ) 1 Vα φ α V α F f (E) γ α (x, y) = (x, φα 1 (f(x), y)) γ α : V α F f (p) 1 (V α ) γ α : V α F 1 F V α V α F 1 Vα f 1 F V α U α F 1 Vα φ 1 α V α p 1 (U α ) p(φ 1 α (f(x), y)) = f(x)

γ α (x, y) f (E) γ α : V α F f (p) 1 (V α ) γ α ψ α (x, e) = γ α (x, φ α (e)) = (x, φ 1 α (f(x), φ α (e))) = (x, φ 1 α (p(e), φ α (e))) = (x, φ 1 α = (x, e) φ α (e)) ψ α γ α (x, y) = ψ α (x, φ 1 α (f(x), y)) = (x, φ α φ 1 α (f(x), y)) = (x, y) γ α ψ α ψ α (x, y) (V α V β ) F ψ β ψα 1 (x, y) ψ β ψ 1 α (x, y) = ψ β γ α (x, y) = ψ β (x, φ 1 α (f(x), y)) = (x, φ β (φ 1 α (f(x), y))) Φ αβ : U α U β (F ) p : E B φ β (φ 1 α (f(x), y)) = Φ αβ (f(x))(y) ψ β ψα 1 (x, y) = (x, Φ αβ (f(x))(y)) G M 0 M 1 M n nπ M n S 1 ξ n φ n : S 1 S 1

z S 1 = {z C z = 1} φ n (z) = z n ξ 1 φ n ξ n p : E B f : X B p (f) : f (E) E x X x x = f (p) 1 (x) = {(x, e) f (E) f (p)(x, e) = x} = {(x, e) X E f(x) = p(e)} = {e E f(x) = p(e)} = p 1 (f(x)) = f(x) (x, e) e p (f) p (f) x f(x) p : E B G f (p) V α V β = f 1 (U α ) f 1 (U β ) f Φ U α U αβ β (F ) p G f (p) V α V β U α U Φαβ β G (µ) (F ) f U α U β Φαβ G (µ) (F ) p (f) p : E B p : E X

E f E p X f B p E p X f (E) f (p) X ( f, f) : (E, X) (E, B) p f = f p e E p( f(e)) = f(p (e)) f (E) (p (e), f(e)) f (E) f : E f (E) f(e) = (p (e), f(e)) p E X f = f (E) X f (p) f E f f (E) p (f) E f p (f) f f {φ λ : p 1 (U λ ) U λ F } λ Λ {φ γ : p 1 (V γ ) V γ F } γ Γ p : E B p : E X {V γ } γ Γ f(v γ ) U λ f (E) X {f 1 (U λ )} λ Λ {ψ λ : p (f) 1 (U λ ) f 1 (U λ ) F } λ Λ G f(v γ ) U λ ( 2 ψ λ f φ γ 1 ) U λ (F ) (µ) G

φ λ (e) = (p(e), ( 2 φ λ )(e)) f (E) ψ λ : f (p) 1 (U λ ) U λ F ψ λ (x, e) = (x, ( 2 φ λ )(e)) V γ F φ 1 λ p 1 f (Uλ ) f (p) 1 (U λ ) ψ λ U λ F (x, y) (x, 2 (φ λ ( f(φ γ 1 (x, y))))) = (x, (2 φ λ f φ γ 1 )(x)(y)) ( 2 ψ λ f φ γ 1 ) = (2 φ λ f φ γ 1 ) f ( 2 φ λ f φ γ 1 ) ( 2 φ λ f φ γ 1 ) U α (F ) (µ) G f X ξ = (p : E B) f : X B g : Y X (f g) (ξ) = g (f (ξ)) ξ = (p : E B) 1 B(ξ) = ξ

p (1 B ) : 1 B(E) E p : E B A B i : A B i (E) E A E A p : E B Y X f g Z X Z Y = {(x, y) X Y f(x) = g(y)} W f Y g X f Z g W f X Z Y g Y f Z g : W X Z Y f Y Z g

ξ B f, g : X B f (ξ) = g (ξ) f g f (ξ) = g (ξ) f g f (ξ) = g (ξ) f 0, f 1 : X Y H : X [0, 1] Y x X H(x, 0) = f 0 (x) H(x, 1) = f 1 (x) f 0 f 1 f 0 f 1 H f 0 f 1 H f 0 f 1 t [0, 1] x X f t : X Y f t (x) = H(x, t) f t = (H)(t) t [0, 1] f t : X Y t 0 1 f 0 f 1 f 0 f 1 f 0 f 1

C C D C f D D C C f(z) dz = f(z) dz C C D p : E Y p : E X ( f 0, f 0 ) : (E, X) (E, Y ) H : X [0, 1] Y x X H(x, 0) = f 0 (x) X H E [0, 1] p 1 [0,1] X [0, 1] H H E Y p e E H(e, 0) = f 0 (e)

E {0} E [0, 1] = E X {0} X [0, 1] = X f 0 f 0 H H H X p : E Y H E = Y F f0 (e) = ( f 0(e), f 0 (e)) H(e, t) = (H(p (e), t), f 0 (e)) E Y E X H X E Y

X W W X W W X u : X [0, 1] u(w ) = 1 u(x W ) = 0 X W W u : X [0, 1] X [0, 1] E [0, 1] X [0, 1] [0, 1] (X, d) {U λ } λ Λ σ > 0 d(a) < σ λ Λ A U λ d(a) = {d(a, a ) a, a A} A τ {U λ } λ Λ X X [0, 1] {V α } α A [0, 1] 0 < 1 l < 2 l < < l 1 l < 1 X U 1,1,..., U n1,1,..., U 1,l,..., U nl,l X

W 1 = [0, 4 3l ) W 2 = ( 2 3l, 7 3l )... W l = ( 3l 4 3l, 1] l nk k=1 i=1 U i,k W k = X [0, 1] (i, k) U i,k W k V α α A (x, t) X [0, 1] (x, t) U x,t V x,t V α α U x,t X V x,t [0, 1] t [0, 1] {U x,t } x X X X U x1,t... U xnt,t X n t i=1 V t = n t i=1 V x i,t U xi,t V t = X V t {V t } t [0,1] [0, 1] [0, 1] σ 5 3l < σ l N W 1 = [0, 4 3l ) W 2 = ( 2 3l, 7 3l )... W l = ( 3l 4 3l, 1] d(w k) 5 3l k W k V tk t k k 1 i n tk n tk i=1 U xi,t k V tk = X V tk [0, 1] X U xi,t k W k U xi,t k V xi,t k V α 0 < 1 l < 2 l < < l 1 l {V α } α A V α E Y Y V α φ α : p 1 (V α ) V α F < 1

{H 1 (V α )} α A X [0, 1] [0, 1] 0 < 1 l < 2 l < < l 1 l < 1 U 1,1,..., U n1,1,..., U 1,l,..., U nl,l X l n k k=1 i=1 U i,k W k = X [0, 1] (i, k) U i,k W k H 1 (V α ) α A W 1 = [0, 4 3l ) W 2 = ( 2 3l, 7 3l )... W l = ( 3l 4 3l, 1] t k = k l t k W k W k+1 k k H E [0, t k ] k = 0 E {0} f 0 E [0, t k ] E [0, t k+1 ] E [t k, t k+1 ] H X [t k, t k+1 ] X x U i,k+1 x O, O O, O U i,k+1 X (O, O) X {(O j, O j)} j=1,...,s X = s j=1 O j [0, 1] [t k, t k+1 ] (O j, O j) u j : X [t k, t k+1 ] u j (O j ) = t k+1 u j (X O j) = t k x X τ 0 (x) = t k τ j (x) = {u 1 (x),, u j (x)} t k = τ 0 (x) τ 1 (x)... τ s (x) = t k+1 X j = {(x, t) X [t k, t k+1 ] t τ j (x)} X [t k, t k+1 ] X 0 = X {t k } X 1 X 2 X s = X [t k, t k+1 ] E [0, 1] X j E j E 0 = E {t k } E 1 E s = E [t k, t k+1 ]

E j 1 H j 1 H j : E j [0, 1] E X j X j 1 1 j s { } X j 1,j = (x, t) O j [t k, t k+1 ] τ j 1 (x) x τ j (x) X j = X j 1 X j 1,j U i,k+1 V α X j 1,j O j [t k, t k+1 ] O j [t k, t k+1 ] U i,k+1 [t k, t k+1 ] H(U i,k+1 [t k, t k+1 ]) V α t k+1 u j τ j 1 X j 1,j t k O j O j X X [0, 1] E j 1,j = E Xj 1,j E j = E j 1 E j 1,j V α E Y φ α (e) = (p(e), φ α (e)) (e, t) E j 1,j H j (e, t) = φ 1 α (H(p (e), t), φ α ( H j 1 (e, τ j 1 (p (e))))) E j 1,j H j E j 1 [0,1] = H j 1

E j 1,j E j 1 t = τ j 1(p (e)) = φ 1 α (H(p (e), t), φ α ( H(e, t))) = φ 1 α (p H(e, t), φ α ( H(e, t))) = φ 1 α = H(e, t) φ α ( H(e, t)) H j 1 H j 1 E j H E [0, 1] H X [0, 1] X A B f A B = g A B f : A Y g : B Y f g : A B Y (f g) A = f (f g) B = g τ j X {U α } α A 1 f γ : X [0, 1] (γ Γ) {(f γ )} γ Γ X X = γ Γ (f γ )

x X x U (f γ ) (f γ ) = {x X f γ (x) 0} x X f γ (x) = 1 γ Γ γ Γ (f γ ) U α α A 1 {U α } α A B p : E B U α p X X B p : E B p : E B f : X B f : X E X 8888888 f 8 f E p B

H : X [0, 1] B f H(x, 0) = f(x) B H : X [0, 1] E H(x, 0) = f(x) 888888888888 E H 8 H X [0, 1] B p X {0} X [0, 1] H f H E B p H E X X = X X X {U α } α A X {V β } β B β B V β U α α A x X x U U V β

B p : E B X X p : E X [0, 1] X p : E X p : E X [0, 1] p 1 [0,1] : E [0, 1] X [0, 1] i 0 : X = X {0} X [0, 1] E = i 0(E) p E i 0 E X X [0, 1] E E p p H H = 1 X [0,1] : X [0, 1] p = X [0, 1] H(x, 0) = (x, 0) = i 0 (x) E [0, 1] H E p 1 [0,1] p H X [0, 1] X [0, 1] H p : E Y f 0, f 1 : X Y f 0 f 1 X f0 (E) = f 1 (E) X = X

H : X [0, 1] Y f 0 f 1 E H H (E) X [0, 1] X E X H (E) = E [0, 1] t [0, 1] i t : X = X {t} X [0, 1] f 0 = H i 0 f 1 = H i 1 f 0 (E) = (H i 0 ) (E) = i 0(H (E)) = i 0(E [0, 1]) f 1 (E) = (H i 1 ) (E) = i 1(H (E)) = i 1(E [0, 1]) t [0, 1] E = E {t} E [0, 1] X = X {t} i t X [0, 1] E = i 1(E [0, 1]) f 0 (E) = i 0(E [0, 1]) = E = i 1 (E [0, 1]) = f 1 (E) X X X x 0 X H : X [0, 1] X

x X H(x, 0) = x H(x, 1) = x 0 X 1 X x 0 c x0 H X x 0 x 0 n D n = {(x 1,, x n ) R n x 2 1 + + x 2 n 1 } R n R n X p : E X F X E X F = E X = X X x 0 X 1 X c x0 H : X [0, 1] X 1 X(E) = c x 0 (E)

1 X(E) = {(x, e) X E 1 X (x) = p(e)} = {(x, e) X E x = p(e)} = {(p(e), e) X E e E} = E c x 0 (E) = {(x, e) X E c x0 (x) = p(e)} = {(x, e) X E x 0 = p(e)} = X {e E x 0 = p(e)} = X p 1 (x 0 ) = X F E = X F n D n S+ n = {(x 0,, x n ) S n x n 0} S n = {(x 0,, x n ) S n x n 0} n S n S n = S n + S n S n + S n D n G p : E S n F i + : S n + S n i : S n S n S n + S n E S n + S n i +(E) i (E) φ + : i +(E) = S n + F φ : i (E) = S n F

(S+ n S ) n F φ 1 + p 1 (S+ n S ) n φ (S+ n S ) n F B(p) p S n 1 = S n + S n (F ) p G S n + S n 0 < ε < 1 U +,ε = {(x 0,, x n ) S n x n ε} U,ε = {(x 0,, x n ) S n x n ε} i ±,ε : U ±,ε S n U ±,ε φ ±,ε : i ±,ε(e) = U ±,ε F Φ + : U +,ε U,ε (F ) φ ±,ε p G G (F ) Φ + Φ + U +,ε U,ε Φ + S n 1 = S+ n S n Φ + U +,ε U,ε G B(p) E S n + G S n G G U + U B(p) G p : E S n p : E S n

G E E 0 < ε < 1 ψ ±,ε : p 1 (U ±,ε ) U ±,ε G ψ ±,ε : (p ) 1 (U ±,ε ) U ±,ε G E E B(p) B(p ) B(p) B(p ) E p B E p B F G G F {F i } i=1,...,n B i F i U i U i E E U i 1 i, j n { Φ ij : F i F j G } { Ψ ij : F i F j G } E E E E 1 i n 1 i, j n x F i F j λ i : F i G λ j (x) 1 Ψ ij (x)λ i (x) = Φ ij (x) α f α : p 1 (U α ) p 1 (U α ) f : E E ( ) B(p) B(p ) f : E E f λ + : U +,ε G λ : U,ε G

B(p) B(p ) E E x S n 1 = S+ n S n B(p)(x) = λ (x) 1 B(p )(x)λ + (x) λ ± B(p) B(p ) S+ n S+ n x 0 = (1, 0,..., 0) S n 1 H + : S+ n [0, 1] S+ n x S+ n H + (x, 0) = x H + (x, 1) = x 0 H + (x, t) S+ n [0, 1] h + (x, t) = λ + (H + (x, t)) S n x 0 H : S n [0, 1] S n h (x, t) = λ (H (x, t)) (x, t) S n 1 [0, 1] F (x, t) = h (x, t) 1 B(p )(x)h + (x, t) F (x, 0) = h (x, 0) 1 B(p )(x)h + (x, 0) = λ (x) 1 B(p )(x)λ + (x) = B(p)(x) F (x, 1) = h (x, 1) 1 B(p )(x)h + (x, 1) = λ (x 0 ) 1 B(p )(x)λ + (x 0 ) B(p) x λ (x 0 ) 1 B(p )(x)λ + (x 0 ) B(p λ (x 0 ) e ) G λ + (x 0 ) e

w : [0, 1] G e G (x, t) w + : [0, 1] G S n 1 [0, 1] G(x, t) = w (t) 1 B(p )(x)w + (t) G(x, 0) = F (x, 1) G(x, 1) = B(p )(x) B(p) B(p ) ( ) B(p) B(p ) x S n 1 L(x) = B(p )(x) 1 B(p)(x) B(p) B(p ) L c e H : S n 1 [0, 1] G c e G e H(S n 1 {1}) = c e (S n 1 ) = {e} H H : S n 1 [0, 1]/S n 1 {1} G S n 1 [0, 1]/S n 1 {1} S n + = S n 1 [0, 1]/S n 1 {1} S n +

H λ + : S n + G λ = c e : U,ε G Φ + Φ + E E x S n + S n λ (x) 1 Φ + (x)λ + (x) = e Φ + (x) H(x, 1) = Φ + (x)l(x) = B(p )(x)b(p )(x) 1 B(p)(x) = B(p)(x) E E G S 1 f : X Y c y0 f f : X [0, 1]/X {1} Y f([x, 1]) = y 0 X = X {0} X [0, 1]/X {1} X X [0, 1]/X {1} X CX X CX CS n 1 D n S n +

D n S n X X {φ λ : D n λ X} λ Λ λ φ λ : D n λ X e λ = φ λ (D n λ ) X = λ Λ e λ e λ e λ = φ λ (D n ) e λ = n r X (r) = µ Λ eµ r X r e λ = n e µ φ λ ( D n ) = φ λ (S n 1 ) X (n 1) {φ λ } λ Λ {e λ } λ Λ X = λ Λ e λ e λ φ λ e λ n n n S 1 S 1 S 2 = e 0 + e 0 e 1 + e 1 e 2 + e 2 S 2 n S n S n = e 0 + e 0 e 1 + e 1 e n + e n S n

e 1 + e 0 e 0 + e 1 S 1 = e 0 + e 0 e 1 + e 1 e 1 e 2 + e 0 e 0 + e 2 e 1 + S 1 0 S n e 0 S n e n = S n {e 0 } e n n S n = e 0 e n S n e 0 S n S n {e 0 } R n (D n ) X

e 1 + e 0 e 0 + e 1 S 1 S n {e 0 } = (D n ) S n n (n 1) X n e n X = X (n 1) e n = X (n 1) e n X (n 1) e n e n e n φ φ S n 1 : D n = S n 1 X (n 1) φ((d n )) = e n X (n 1) X D n φ S n 1 X (n 1) n n 0 (n 1) n

D n = φ S n 1 X n 1 X Y A Y f : A X f Y X X f Y X f Y = (X Y )/ f f a A a f f(a) X f Y X Y = D n A = D n = S n 1 f : S n 1 X f Y = f D n = D n / D n S n n φ n : D n S n φ n ( D n ) = e 0 φ n : D n / D n S n

D n / D n S n X X = e λ λ Λ φ λ : D n X e λ n X (n) = X (n 1) φλ n 1 S Dλ n λ e λ =n D n λ Dn X (n 1) X (n) n φ λ X (n 1) φλ S n 1 X (n) e λ =n X (n) X X 0 X = {x} x X x X D0 x X X = λ Λ e λ e λ e λ D n λ A X X A e λ λ Λ e λ A X : A e λ e λ

C R (n 1) n X X = e λ λ Λ φ λ : D n X e λ n 0 p n : X (n 1) φλ S n 1 e λ =n D n λ X (n) X (n 1) n p n p : X (n 1) A X (n 1) φλ S n 1 X (n 1) φλ S n 1 Dλ n Dλ n e λ =n e λ =n e λ =n D n λ : p 1 (A) : ( ) p 1 (A) = (X (n 1) A) D n λ p 1 (A) A X (n 1) A Dλ n p 1 (A) p n (A) X (n) X (n) X n e µ p n (A) e µ e µ ( p n (A) ē µ = p n (X (n 1) A) ) p n (D λ p 1 (A)) ē µ λ

ē µ e λ1,..., e λk p n (A) ē µ = = ( ) p n (X (n 1) A) p n (Dλ n 1 p 1 (A)) p n (Dλ n k p 1 (A)) ē µ ( ) p n (X (n 1) A) φ λ1 (Dλ n 1 p 1 (A)) φ λk (Dλ n k p 1 (A)) ē µ p n X (n 1) X (n 1) p n (X (n 1) A) φ λi (D n λ i p 1 (A)) p n (A) ē µ X φ λ : D n ē λ D n X φ λ φ λ : D n e λ X n X (n) X X (n) X S n X { e λ e λ X : }, X =, X X X X A X X A X

G G X G P G (X) G X P G (X) X X Y (X, Y ) [X, Y ] X Y G BG G p : EG BG X [f] f (EG) B : [X, BG] P G (X) X G E X f : X BG f (EG) = E E f (EG) EG X = f f, g f g G G p : EG BG BG

D n = S n 1 X n n k n k X [S k, X] = { } k n f, g : S k X f g 0 X 0 X 0 x 0, x 1 X f : S 0 X g : S 0 X f(1) = x 0, f( 1) = x 0 g(1) = x 0, g( 1) = x 1 S 0 = { x R x 2 = 1 } = {1, 1} f g X 0 f g H : S 0 [0, 1] X

f g w(t) = H( 1, t) w(0) = H( 1, 0) = f( 1) = x 0 w(1) = H( 1, 1) = g( 1) = x 1 w x 0 x 1 X X 0 X x 0 X (X, x 0 ) x 0 X (X, x 0 ) (Y, y 0 ) (X, x 0 ) (Y, y 0 ) f : X Y f(x 0 ) = y 0 f : (X, x 0 ) (Y, y 0 ) g : (X, x 0 ) (Y, y 0 ) f g f g H : X [0, 1] Y t [0, 1] H(x 0, t) = y 0 f g X Y ((X, x 0 ), (Y, y 0 )) (X, Y )

[(X, x 0 ), (Y, y 0 )] = [X, Y ] = (X, Y )/ X Y X = S n e 0 = (1, 0,..., 0) R n+1 S n (X, x 0 ) π n (X, x 0 ) = [(S n, e 0 ), (X, x 0 )] n 1 X n π n (X) n (X, x 0 ) X n k n [S k, X] = { } k n π k (X, x 0 ) = { } k n f : S k X F S k = f F : D k+1 X f : S k X D k+1 k n x 0 X c x0 : S k X x 0 X n f : S k X (X, Y )

c x0 H : S k [0, 1] X c x0 f H(S k {0}) = c x0 (S k ) = {x 0 } H H : S k [0, 1]/S k {0} X S k [0, 1]/S k {0} = D k+1 H F S k = f F : D k+1 X = S k [0, 1]/S k {1} D k+1 f, g : S k X f g f F : D k+1 X p : S k [0, 1] S k [0, 1]/S k {0} H : S k [0, 1] X x S k S k [0, 1] p S k [0, 1]/S k {0} = D k+1 F X H(x, 0) = F (0) H(x, 1) = F S k(x) = f(x)

H(x, 0) x 0 = F (0) H f c x0 g g c x1 G h : S 0 X h(1) = x 0 h( 1) = x 1 h : D 1 = [0, 1] X h c x0 c x1 H G f g e 0 S n S n [0, 1]/S n {0} {e 0 } [0, 1] = D n+1 (X, x 0 ) X [0, 1]/(X {1} {x 0 } [0, 1]) X CX X n 1 π n (X, x 0 ) X Y X Y X Y X Y x 0 y 0 X Y X Y X Y = X {y 0 } {x 0 } Y X Y k 1 : S k S k S k

X Y S k S k S k E = { (x 0,..., x k ) S k x k = 0 } S k X : X X X X x, x = x (x, x 0 ) = x, x = x 0 X f, g : S k X f + g : S k X

S k S k S k f g X X X S k S k S k + X A (X, A) (Y, B) f : X Y f(a) B f f : (X, A) (Y, B) g : (X, A) (Y, B) f g f g H : X [0, 1] Y a A t [0, 1] H(a, t) B (X, A) (Y, B) ((X, A), (Y, B)) [(X, A), (Y, B)] A B

(X, x 0 ) [([0, 1] n, [0, 1] n ), (X, x 0 )] = π n (X, x 0 ) R n R n { } τ : [0, 1] n / [0, 1] n R n { } ( 2πx 1 1 2,..., 2πx ) n 1 2, (x1,..., x n ) [0, 1] n τ([x 1,..., x n ]) =, (x 1,..., x n ) [0, 1] n π : (([0, 1] n, [0, 1] n ), (X, x 0 )) (S n, X) f : ([0, 1] n, [0, 1] n ) (X, x 0 ) f( [0, 1] n ) = {x 0 } f : [0, 1] n / [0, 1] n X S n = R n { } S n = [0, 1] n / [0, 1] n f π(f) : S n X π : [([0, 1] n, [0, 1] n ), (X, x 0 )] π n (X, x 0 ) f : S n X [0, 1] n [0, 1] n / [0, 1] n = S n f X ([0, 1] n, [0, 1] n ) (X, x 0 ) π [f], [g] [([0, 1] n, [0, 1] n ), (X, x 0 )] π([f]) = π([g]) f ḡ H [0, 1] n [0, 1] n / [0, 1] n H X f g [f] = [g]

h : (X, x 0 ) (Y, y 0 ) [([0, 1] n, [0, 1] n ), (X, x 0 )] π n (X, x 0 ) π h [([0, 1] n, [0, 1] n ), (X, x 0 )] π n (Y, y 0 ) h h π n (X, x 0 ) + f, g : ([0, 1] n, [0, 1] n ) (X, x 0 ) f(t 1,..., t n 1, 2t n ), 0 t n 1 2 (f + g)(t 1,..., t n ) = 1 g(t 1,..., t n 1, 2t n 1), 2 t n 1 φ : [0, 1] n / [0, 1] n = S n ( n ) i=1 2 2πt i π 2 1, 2πt 2 1 π n i=1 φ([t 1,..., t n ]) = 2 2πt i π 2,..., 2πtn π 2 2 +1 n i=1 2 2πt i π 2 2 +1 n, (t i=1 2 2πt i π 1,..., t n ) [0 2 +1 e 0, (t 1,..., t n ) [0 ((x 0,..., x n 1, 1 2x n ), e 0 ), 0 x n 1 (x 0,..., x n ) = (e 0, (x 0,..., x n 1, 2x n + 1)), 1 x n 0 k 1 + + : π k (X) π k (X) π k (X) f 0, f 1, g 0, g 1 : S k X π h f 0 f 1, g 0 g 1 f 0 + g 0 f 1 + g 1 k 1 π k (X, x 0 ) : S k X

k 2 π k (X, x 0 ) 0 k 2 π k (X) f, g : S k X f + g g + f t [0, 1] F t : S k S k S k e 0 e 0 l πt t F : S k [0, 1] S k 1 S k l π H : S k [0, 1] F S k f+g X H Sk {0} = f + g H S k {1} = g + f H f + g g + f X Y f : X Y g : X Y g X n Y n g(x (n) ) Y (n) f g

X Y f : X Y n 0 f(x (n) ) Y (n) f S n (n 1) k n 1 π k (S n ) = 0 k n 1 f : S k S n S k S n S n S n = { } (S n { }) = e 0 e n f g g : S k S n k n 1 g(s k k) = g(s k ) S n k = e 0 = { } g f : S k S n X A p : E Y f : X Y f : X E

X 8888888 f 8 f E p Y H : X [0, 1] Y f H(x, 0) = f(x) G : A [0, 1] E G A {0} = f A A [0, 1] X [0, 1] G H E Y p H : X [0, 1] E A [0, 1] X {0} G H E p X [0, 1] H Y H : X [0, 1] E X {0} A [0, 1] f G E p X [0, 1] H Y

G : A [0, 1] E X X X X Y p : E Y f : X Y f : X E X 8888888 f 8 f E p Y H : X [0, 1] Y f x X t [0, 1] H(x, 0) = f(x) H(, t) = H : X [0, 1] E H(x, 0) = f(x) 888888888888 E H 8 H X [0, 1] Y p A = { }

G EG BG B : [X, BG] P G (X) G EG BG G G p : E B [X, B] P G (X) G E B E (n 1) G X X < n f f (E) [X, B] P G (X) E X X G f : X B f (E) = E X f r = 0, 1, 2,..., n 1 i r : X (r) X f r : X (r) B f r : i r(e ) = f r (E)

X (r 1) i r 1 r X (r) f r 1 f r B B r f r f r : i r(e ) E r = 0 X 0 X (0) X 0 0 X (0) 0 X (0) f 0 f 0 (E) = X (0) G = i 0(E ) f r 1 X (r) = X (r 1) (r) r f r f r 1 e r X r φ : D r X e r D r φ (E ) j : S r 1 D r φ (E ) = D r G j φ (E ) = (φ j) (E ) = S r 1 G S r 1 G = j φ (E ) i r 1(E ) = f r 1 (E) E S r 1 = S r 1 φ j X (r 1) = X (r 1) f r 1 f r 1 : S r 1 G E B

h : S r 1 E h(x) = f r 1 (x, e) e G E (n 1) r < n h h : D r E p : E B G G E µ : E G E F (x, g) = µ( h(x), g) F : D r G E F : D r B F (x) = p F (x, e) D r G D r F F p E B f r : X (r) B r e r x e r x = φ(y) y D r f r (x) = F (y) X (r 1) f r 1 h h D r F S r 1 f r 1 y f r : X (r) B F f r i r(e ) E i r(e ) = f r (E)

f = f n 1 f (E) = f n 1(E) = i n 1(E ) = E f f (E) [X, B] P G (X) X X (0) X [X, B] P G (X) f (E) = g (E) f g H : X [0, 1] B X [0, 1] [0, 1] = {0} (0, 1) {1} [0, 1] X Y X Y X Y X Y X X Y X X Y G p : E B E n G X X < n f f (E) [X, B] P G (X)

f, g : X B f (E) = g (E) f g H : X [0, 1] B E = f (E) f (p) 1 [0,1] : E [0, 1] X [0, 1] X {0} f (E) X {1} g (E) X [0, 1] G X < n X [0, 1] X [0, 1] < n + 1 E n H : X [0, 1] B H (E) = E [0, 1] f g : X {0} X {1} B H H f g G G E B n n E (n 1) G n n G G G G G G G G

p : E B E B F E B p : E B f : X Y f : π k (X) π k (Y ) [g] π k (X) f ([g]) = [f g] [g] g : S k X f f k 1 f f : X Y g : Y Z (g f) = g f 1 X : X X (1 X ) = 1 πk (X) π k (E) π k (B) p : π k (E) π k (B) V W f : V W f = f 1 (0) f = W / f

f f = 0 f = 0 f : G H f G H f = {g G f(g) = e} f = H/ f f H f f f f 1 f 2 G 1 G2 G3 G 2 f 1 = f 2 f 2 f 1 = 0 f 2 (x) = 0 y G 1, x = f 1 (y) f k 1 f k G k 1 Gk Gk+1 k f k 1 G k 1 Gk f k Gk+1 G k

F p : E B B i : F = p 1 ( ) E n 1 π n (F ) i π n (E) p π n (B) p i = 0 p ([f]) = 0 [g] π n (F ), i ([g]) = [f] F = p 1 ( ) p i(f ) = { } p i = p i = 0 [f] π n (E) p ([f]) = 0 p f H : S n [0, 1] B H Sn {0} = p f S n 88888888 f 8 p f E B p H : S n I E p H = H H Sn {0} = f

g = H Sn {1} p(g(s n )) = { } p g = p H Sn {1} = H Sn {1} = g(s n ) p 1 ({ }) = F [g] π n (F ) H f i g [f] = i ([g]) g : S n F n π n (F ) i π n (E) p π n (B) p : E B i : F E p i π n (S n ) = Z π n (D n+1 ) = 0 D n+1 S n = D n+1 i : S n D n+1 i : π n (S n ) = Z 0 = π n (D n+1 ) F i E p B i : π n (F ) π n (E)

i [f] π n (F ) i ([f]) = 0 i ([f]) = 0 f E f : D n+1 E f D n+1 = f f : S n F (p f)( S n ) = { } p f f : D n+1 / D n+1 = S n+1 B i π n+1 (B) p : E B F B n 1 : π n (B) π n 1 (F ) [f] π n (B) f f : S n B S n = D n / D n f : D n B f( D n ) = { } D n = S n 1 [0, 1]/S n 1 {0} { } [0, 1] π : S n 1 [0, 1] S n 1 [0, 1]/S n 1 {0} { } [0, 1] = D n

x S n 1, t [0, 1] H : S n 1 [0, 1] π D n f B H(x, 0) = H(, t) = H H : S n 1 [0, 1] E p H = H H S n 1 {0} = g = H S n 1 {1} : S n 1 E p g = p H S n 1 {1} = H S n 1 {1} = f D n = g(s n 1 ) p 1 ( ) = F [g] π n 1 (F ) : π n (B) π n 1 (F ) ([f]) = [g] H n 2 i n 2 π n (B) π n 1 (F ) i π n 1 (E)

H : S n 1 I E g E i ([f]) = i ([g]) = 0 i = 0 [g] π n 1 (F ) i ([g]) = 0 g E G : S n 1 [0, 1] E G = p G : S n 1 [0, 1] B H S n 1 {0} = H S n 1 {0} = H H H f : D n = S n 1 [0, 1]/S n 1 {0} { } [0, 1] B f S n 1 = H S n 1 {1} = p g = f π n (B) [f] = [g] n 2 π n (E) p π n (B) π n 1 (F ) p = 0 [f] π n (E) f : D n E f( D n ) = { } p f H S n 1 [0, 1] π D n f E

([p f]) = [ H S n 1 {1}] = [f S n 1] = [ ] = 0 p = 0 [f] π n (B) ([f]) = 0 H : S n 1 [0, 1] E S n 1 [0, 1] π D n f B F H S n 1 {1} G : S n 1 {1} [0, 1] F E G : (S n 1 {0} { } [0, 1]) [0, 1] E H : S n 1 [0, 1] {0} E G G : (S n 1 {0, 1} { } [0, 1]) [0, 1] E X = S n 1 [0, 1] A = S n 1 {0, 1} { } [0, 1] K = G G H A {0} = K A {0} H : X [0, 1] B H(x, t, s) = p H(x, t) X {0} A [0, 1] H K E X [0, 1] H B p

K H K : X [0, 1] E K(S n 1 {0} { } [0, 1]) = { } K : D n [0, 1] = (S n 1 [0, 1]/S n 1 {0} { } [0, 1]) [0, 1] E g = K D n {1} [g] π n (E) p [g] = [f] g(s n 1 ) = K(S n 1 {1} {1}) = { } π n (F ) i π n (E) p π n (B) π n 1 (F ) π 1 (B) E B F [l] π 1 (B) l p : π 1 (E) π 1 (B) l : [0, 1] B l(0) = l(1) = [0, 1] = { } [0, 1] l B x 0 p 1 ( ) { } {0} E x 0 { } {0} { } I l E B p

{ } {0} { } I l l E B p l : [0, 1] E ω l x 0 l l(1) = x 0 F F = p 1 ( ) l(0) = x 0 l(1) [ˆl] π 1 (E) ω : [0, 1] E l(2t) 0 t ˆl(t) 1 2 = 1 ω(2t 1) 2 t 1 p ([ˆl]) = [l] f : G H 0 f G f H 0

π n (F ) i π n (E) p π n (B) π n 1 (F ) π 1 (B) 0 E B p : E B G π 0 (G) G π n (G) i π n (E) p π n (B) π n 1 (G) π 1 (B) π 0 (G) E π n (G) i π n (E) p π n (B) π n 1 (G) π 1 (B) π 0 (G) 0 F E B π 1 (B) π 0 (F ) π 0 (E) π 0 (B) G G EG BG EG EG 0 EG G EG EG/G G BG = EG/G EG G { }

EG S n (n 1) n S { } S = (x 0, x 1,...) x i R, 0, x 2 i = 1 S n = { (x 0, x 1,..., x n ) R n+1 x 2 0 + + x 2 n = 1 } i=0 i n : S n S n+1 i n (x 0,..., x n ) = (x 0,..., x n, 0) i n S n S n+1 S n S n+1 S S = S A S : A S n S n X {X α } α A X X A X : A X α X α n=1 S n

X {X α } α A X = A X : A X α X α X X {X α } α A α A S = n=1 X α S S n S X {X n } X X n X n+1 X = n=1x n X {X n } K X K X n n K X n K (X n X n 1 ) ϕ n n {n 1, n 2, } K (X nk X nk 1) x k A = {x 1, x 2, } n A X n X A X A K K A A i = A {x i } {x i }, n = n i A i X n = ϕ, n {n 1, n 2,...}

A i {x i } A A A = {x i } i=1 A i π i (S ) = 0 S f : S i S f(s i ) S n f(s i ) S n n > i S n f H : S i I S n S i I H S n S f S G S G G 2 C 2 S n R n = S n /C 2 n S n R n C 2 S n S n+1 R n R n+1

R = R n n=1 C 2 S R S R C 2 X [X, R ] = P C2 (X) H (X; Z/2Z) X Z/2Z H 1 (X; Z/2Z) = [X, R ] X C 2 X 1 2 C 2 ξ = (E p X) P C2 (X) = [X, R ] = H 1 (X; Z/2Z) H 1 (X; Z/2Z) w 1 (ξ) ξ C 2 G C 2 S S C 2 S S

S S n (n 1) C 2 S n S n = O(n + 1)/O(n) S n O(n + 1) O(n) O(n) S n (n 1) O(n) O(n + 1) O(n) O(n + k) A A 0 0 I k O(n) O(n + k) n,k (R) = O(n + k)/o(n) n,k (R) (n 1) G H G G/H H K G/K G/H H/K G G/H HH H H/K µ : H H/K H/K

µ(h, h K) = hh K H/K G H H/K G/H G/K G/H G H H/K G/K (g, hk) ghk G H H/K G/K G/H = G/H k k = 1 n,1 (R) = S n n n,k (n 1) n,k+1 (R) (n 1) O(n + k + 1)/O(n) O(n + k + 1)/O(n + 1) O(n + 1)/O(n) S n n,k+1 (R) n+1,k (R) π i (S n ) π i ( n,k+1 (R)) π i ( n+1,k (R)) i < n π i (S n ) = 0 π i ( n+1,k (R)) = 0

i < n 0 π i ( n,k+1 (R)) 0 i < n π i ( n,k+1 (R)) = 0 0 G 0 G = 0 n O(n + k) O(n + k + 1) A 1 0 0 A n,k (R) = O(n + k)/o(n) O(n + k + 1)/O(n + 1) = n+1,k (R) E(k) = n,k (R) n=1 E(k) E(k) C 2 = O(1) S = E(1) O(k) E(k) S S n R n+1 C 2 E(k) (A, B) A 0 0 B O(n) O(k) O(n + k) n,k (R) = O(n + k)/o(n) O(k)

n,k (R) = O(n + k)/o(n) O(n + k)/o(n) O(k) = n,k (R) O(n) O(k)/O(n) = O(k) O(k) n,k (R) n,k (R) A 1 n 0 0 A O(k) O(n + k) O(k) O(n + k) O(n) O(k) O(n + k) O(n + k) O(k) O(n + k)/o(n) O(n + k)/o(n) O(k) O(n + k) O(n + k + 1) n,k (R) n+1,k (R) B(k) = n,k (R) n=1 E(k) B(k) O(k) O(k)

X [X, B(k)] = P O(k) (X) O(k) O(k) O(1) = C 2 O(k) = 1 2k(k 1) O(1) = 0 O(k) G G O(k) BG = O(n + k)/o(n) G = E(k)/G n=1 E(k) BG G G O(k) X [X, BG] = P G (X) O(k) G G e U U G {e} G O(k) O(n + k)

O(k) G G O(k) Z Z Z f : R S 1 f(x) = e 2πix Z R f : R S 1 Z f : R S 1 Z G G X Y X Y X Y X Y = X I Y / x, x X, y, y Y (x, 0, y) (x, 0, y) (x, 1, y) (x, 1, y )

X 1,..., X n X 1 X 2 X n = ( ((X 1 X 2 ) X 3 ) ) X n X 1,..., X n X 1 X 2 X n = {(t1, x 1,..., t n, x n ) I X I X t 1 + + t n = 1} / (t 1, x 1,..., 0, x i,..., t n, x n ) (t 1, x 1,..., 0, x i,..., t n, x n ) X X (n + 1) (n 1) i < n π i (X } {{ X } ) = 0 n+1 (n 1) G G G X G X = G

G n E n G = G G }{{} n+1 G E n G µ : G E n G E n G µ(g; t 1, g 1, t 2, g 2,..., t n+1, g n+1 ) = (t 1, gg 1, t 2, gg 2,..., t n+1, gg n+1 ) B n G = E n G/G n = BG = B G EG = E G E n G (n 1) p n : E n G B n G G n E n G, B n G X 1,..., X n 1 i n i t i : X 1 X n [0, 1] t i ([s 1, x 1,..., s n, x n ]) = s i a i : t 1 i (0, 1] X i a i ([s 1, x 1,..., s n, x n ]) = x i X 1 X n t i, a i