0 Intoduction 0.1 (localization fomula) T = U(1) M µ M T µ = M M T µ eff M T 2. M T M T Gauss µ µ eff (1) (2) Atiyah-Singe U(1) [At85]

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1 v /02/11 12 Contents 0 Intoduction U(1) Boel U(1) U(1) Thhe Lie the femion side the boson side Boel A 27 A A A

2 0 Intoduction 0.1 (localization fomula) T = U(1) M µ M T µ = M M T µ eff M T 2. M T M T Gauss µ µ eff (1) (2) Atiyah-Singe U(1) [At85] de Rham [BT82] U(1) Boel U(1) U(1) U(1) 2

3 0.2.3 Thhe (classifying space) (equivaiant cohomology) Koszul duality boson-femion Koszul duality ADHM 1 U(1) Boel Lie G M C G M (g, x) gx M g 1 (g 2 x) = (g 1 g 2 )x M G G G G x = {gx g G} M x M G G M/G = {G x x M} G G x = {x} x M M G G x M G M T x M G G T x M = T x M G ν x T x M G ν x T x M G ν x x M G g gx M G x fee G Lie fee M/G M M/G G 3

4 1.1.2 T = U(1) M fee M/T β π : M M/T π β π β = 0. M M T = ø ) dim M/T < dim M T = U(1) = {t C t = 1} 2k + 1 k S 2k+1 = {(z 0, z 1,..., z k ) C k+1 z a 2 = 1} t (z 0, z 1,..., z k ) = (tz 0, tz 1,..., tz k ). fee CP k = {C k+1 1 } a= T = U(1) n M fee T = U(1) S 2k+1 M t (z, x) = (t 1 z, t x) M T (k) = (S 2k+1 M)/T CP k M T = U(1) S 2k+1 fee M T (k) n µ M n < 2k + 1 µ = 0. M T = ø M 4

5 1.1.5 T = U(1) M fee M T (k) M/T S 2k+1 Gysin H n (M T (k), R) = H n (M/T, R) dim M/T < dim M = n H n (M/T, R) = 0 de Rham µ = dν Stokes µ = dν = 0 M M M S 2k+1 M T M T (k) T = U(1) S 2k+1 fee S 2k+1 S 2k T = U(1) n M M M T fee semifee T T T x µ M T (k) n n < 2k + 1 (M M T ) T (k) µ = dν M T M T ε N(ε) ρ : M [0, 1] N(ε) N(ε/2) 1 d(ρν) M N(ε) (1 ρ)ν M µ = M dν = M M T M d(ρν) + d((1 ρ)ν) = d(ρν) = d(ρν). M N(ε) M semifee multiple obit obifold R U(1) semifee 5

6 1.1.8 µ M T (k) n + p n + p < 2k + 1 M µ [BT82] CP k p [ ] [ ] µ = d(ρν) M N(ε) H p (CP k, R) U(1) π : S 2k+1 CP k Gysin H p (CP k ) ɛ H p+2 (CP k ) π Hp+2 (S 2k+1 ) π! H p+1 (CP k ) H (CP k, R) = R[ɛ]/(ɛ k+1 ), deg ɛ = Thom fom Eule fom n M q X X N X i : X N X π : N X X π i = id X N X X ε q π : E X Thom fom E q τ π! (τ) = 1 Thom fom de Rham H cv π! : H q+k cv (E) = H k (X) Thom fom Thom [BT82] E = N X Thom fom τ X N X X M n q σ σ τ X = σ. τ X X Poincaé dual M X Thom fom τ X X e X = i τ X X M Eule fom X q 6

7 i, π X N X τ X π i τ X = τ X π e X N X n q σ σ π e X = σ M X X = M T e X µ = d(ρν) = M N(ε) M T i d(ρν) e M T i µ = H p (CP k, R) M e T M T e X U(1) Boel k U(1) U(1) S 1 S 3 S 2k+1 S 2k+3 CP 0 CP 1 M M M T (0) CP 0 CP k CP k+1 M T (k) M T (k + 1) CP k CP k+1 M T = M T ( ) = k M T (k) CP = k CP k M T = U(1) M M T Boel CP Gysin H (CP, R) = R[ɛ], deg ɛ = 2 H p T (M, R) Boel = H p (M T, R) Boel M T = U(1) H (CP, R) = R[ɛ] H T (M, R) Boel = lim H (M T (k), R). k 7

8 R[ɛ] ɛ 1 e M T T = U(1) M M T = M T ( ) µ i µ µ = R[ɛ]. M M e T M T ɛ 1 e M T e M T M T = M T ( ) M T = M T ( ) M T (k) k M T (k) T = U(1) S 2k+1 M M T (k) M T (k) S 2k+1 M Lie G π : P B P B 2 U(1) M (x a ; a = 1,..., dim M) dx a dx b dx a = dx a dx b, dx a dx a = 0 dx a p p (p-fom) P = (P ab ) a, b=1,..., dim M P a = b P abdx b P 1 P n = (det P )dx 1 dx n 8

9 2.1.2 M Ω p (M) = {p-foms on M} d = d p : Ω p (M) Ω p+1 (M) 1 f df = a f x a dxa p-fom σ = fdx a1 dx ap dσ = (df)dx a1 dx ap = a a i f x a dxa dx a1 dx ap σ 1 x a dx a / x a 1 M = R 3 d gad, ot, div df = f x 1 dx1 + f x 2 dx2 + f x 3 dx3, d ( g 1 dx 1 + g 2 dx 2 + g 3 dx 3) = g 1 x 2 dx2 dx 1 + g 1 x 3 dx3 dx 1 + g 2 x 1 dx1 dx 2 + g 2 x 3 dx3 dx 2 + g 3 x 1 dx1 dx 3 + g 3 x 2 dx2 dx 3 ( g3 = x 2 g ) ( 2 x 3 dx 2 dx 3 g1 + x 3 g ) ( 3 x 1 dx 3 dx 1 g2 + x 1 g ) 1 x 2 dx 1 dx 2, d ( h 1 dx 2 dx 3 + h 2 dx 3 dx 1 + h 3 dx 1 dx 2) = h 1 x 1 dx1 dx 2 dx 3 + h 2 x 2 dx2 dx 3 dx 1 + h 3 x 3 dx3 dx 1 dx 2 ( h1 = x 1 + h 2 x 2 + h ) 3 x 3 dx 1 dx 2 dx 3. dd = 0 20 x a x b = 9 x b x a

10 dd = 0 dσ dσ = 0 σ dd = 0 Poincaé dz {z C z 0} z H p (M) de Rham = Ke d p /Im d p 1 de Rham Poincaé M R Čech 1-fom 1-fom Lie M ξ Lie L ξ ι ξ L ξ : Ω p (M) Ω p (M), ι ξ : Ω p (M) Ω p 1 (M). Lie L ξ ξ paiing 1 ξ = g x a, a a 1,..., a p ι ξ (fdx a1 dx ap ) = 0, ι ξ (fdx a dx a1 dx ap ) = gfdx a1 dx ap. Lie H. Catan L ξ = dι ξ + ι ξ d = (d + ι ξ ) 2 Lie 10

11 2.2.2 d + ι ξ Ω p (M) ξ = {σ Ω p (M) L ξ σ = 0} H. Catan (d + ι ξ ) 2 = 0 on Ω (M) ξ. d + ι ξ Ω ξ(m) = R[ɛ] Ω (M) ξ = i 0 ɛ i Ω (M) ξ ξ 1 R[ɛ] Ω ξ (M) R[ɛ] d ξ = d ɛ ι ξ deg ɛ = 2 d ξ 1 H. Catan d ξ d ξ = dd ɛ(dι ξ + ι ξ d) + ɛ 2 ι ξ ι ξ = 0. d ξ H p ξ (M) ξ de Rham R[ɛ] µ d ξ µ d ξ µ = 0 µ µ p µ p = 2k, 2k + 1 µ = µ (p) + ɛµ (p 2) + ɛ 2 µ (p 4) + + ɛ k µ (p 2k), µ (i) Ω i (M) ξ d ξ µ = dµ (p) + ɛ( ι ξ µ (p) + dµ (p 2) ) + ɛ 2 ( ι ξ µ (p 2) + dµ (p 4) ) + ɛ k+1 ι ξ µ (p 2k). ξ = 0 R[ɛ] H ξ=0(m) = R[ɛ] H (M) de Rham. 11

12 2.2.3 M M R[ɛ] : Ω ξ(m) R[ɛ] M dim M = p 2k + 1, µ Ω p ξ (M) Stokes d ξ µ = ɛ k dµ (p 2k) = 0. R[ɛ] H ξ (M) R[ɛ] M M M ξ 1,..., ξ Ω ξ (M) Ω p (M) ξ = {σ Ω p (M) L ξa σ = 0, a = 1,..., } Ω ξ(m) = R[ɛ 1,..., ɛ ] Ω (M) ξ, deg ɛ a = 2 d ξ = d ɛ a ι ξa ξ 1,..., ξ Ω ξ (M) R[ɛ1,..., ɛ ] H. Catan a=1 d ξ d ξ = 0. d ξ H p ξ (M) R[ɛ1,..., ɛ ] M R[ɛ 1,..., ɛ ] : Ω ξ(m) R[ɛ 1,..., ɛ ], M : H ξ (M) R[ɛ 1,..., ɛ ] M Lie Lie G M G Lie g M Lie ξ g ξ G M σ G ξ g L ξ σ = 0 12

13 2.2.6 The U(1) Catan model T = U(1) Lie t ɛ : t R ξ t ɛ(ξ) = 1 T = U(1) ξ = ɛ T = U(1) M Ω p (M) t = {σ Ω p (M) L η σ = 0 fo any η t} Ω p (M) t = Ω p (M) ξ ξ U(1) Ω T (M) = R[ɛ] Ω (M) ξ = St Ω (M) t d T = d ξ St = R[ɛ] St Ω (M) t ɛ d T d T = 0 (Ω T (M), d T ) U(1) Catan model H p T (M) Catan Catan model U(1) de Rham H p T (M) Catan = H p T (M, R) Boel. St H T (M) Catan = St H (M) de Rham. 2.3 U(1) Basic foms Lie G π : P B P B π : P B π : Ω p (B) Ω p (P ) B G 13

14 π : P B P = R R, B = R, π(x, y) = x, G = R P f(x, y)dx + g(x, y)dy B g = 0, f y = 0 P σ ξ g ι ξ σ = 0 B g = 0 G = R f y = 0, g y = 0 G G basic fom Ω p (P ) basic = {basic p-foms on P } = {σ Ω p (P ) ξ g; ι ξ σ = 0, L ξ σ = 0} = {σ Ω p (P ) ξ g; ι ξ σ = 0, ι ξ dσ = 0} π (Ω p (B)) = Ω p (P ) basic U(1) T = U(1) π : P B θ P T Lie t t Ω 1 (P ) θ P : t Ω 1 (P ) ɛ t, ξ t ι ξ θ P (ɛ) = ɛ(ξ), L ξ θ P (ɛ) = 0 U(1) T = U(1) p P T p P θ P (ɛ) p = 0 14

15 2.3.3 S 2k+1 U(1) T = U(1) Lie t ɛ : t R nomalization t = e i ɛ t T = U(1) C T = U(1) 2k + 1 k S = S 2k+1 = {(z 0, z 1,..., z k ) C k+1 z a 2 = 1} a=0 t (z 0, z 1,..., z k ) = (tz 0, tz 1,..., tz k ). t ɛ S = S2k+1 z a = x a + i y a ɛ = ( y a x a + ) xa y a a ) ɛ (xb + i y b ) = ɛ e i ɛ (x b + i y b ) = i (x b + i y b ) = y b + i x b. ɛ=0 y a x a + xa y a ( y a x a + ) xa y a z b 2 = y a x a + x a y a = 0 S = S 2k+1 ( y a x a + ) xa y a (x b + i y b ) = y b + i x b. a b ( ) θ S (ɛ) = Im z a dz a = ( y a dx a + x a dy a ) a a ι / ɛ θ S (ɛ) = (( y a ) 2 + (x a ) 2 ) = 1. a ξ t ι ξ θ S (ɛ) = ɛ(ξ). θ S Lie t t Ω 1 (S 2k+1 ) dθ S (ɛ) = 2 a dx a dy a 15

16 ι / ɛ dθ S (ɛ) = 2 a ( y a dy a x a dx a ) = d a ((x a ) 2 + (y a ) 2 ) = 0. ξ t L ξ θ S (ɛ) = ι ξ dθ S (ɛ) = 0. θ S U(1) S 2k+1 CP k U(1) T = U(1) π : P B H p T (P ) Catan = H p (B). St t ) θ P H p (P ) basic H p T (P ) Catan P p µ = µ (p) + ɛµ (p 2) + ɛ 2 µ (p 4) + + ɛ k µ (p 2k), µ (i) Ω i (P ) t d T µ = 0 ι ξ µ (p 2k) = 0 µ + d T (ɛ k 1 θ P µ (p 2k) ) = µ + d(ɛ k 1 θ P µ (p 2k) ) ɛ k µ (p 2k) ɛ k 1 µ cohomologous ˆµ (p) Ω p (P ) t d T ˆµ (p) = 0 dˆµ (p) = 0, ι ξ ˆµ (p) = 0. p 1 ν = ν (p 1) + ɛν (p 3) + ɛ 2 ν (p 5) + + ɛ k ν (p 2k 1), ν (i) Ω i (P ) t d T ν = dν (p 1) + ɛ( ι ξ ν (p 1) + dν (p 3) ) + ɛ k+1 ι ξ ν (p 2k 1) Ω p (M) t ι ξ ν (p 2k 1) = 0. 16

17 ν ν + d T (ɛ k 1 θ P ν (p 2k 1) ) = ν + d(ɛ k 1 θ P ν (p 2k 1) ) ɛ k ν (p 2k 1) ɛ k 1 d T ν ν Ω p 1 (P ) t d T ν Ω p (P ) t ι ξ ν = 0. H p T (P ) Catan = H p (P ) basic = H p (B) T = U(1) π : P B P µ Ω (P ) t 0: µ = 0 R[ɛ]. P ) dim B < dim P de Rham Ω c de Rham H c Thom H p c (R n ) = { R p = n 0 p n R n : H n c (R n ) = R T = U(1) de Rham H T, c T = U(1) R n H n T, c(r n ) = H n c (R n ) R[ɛ] = R[ɛ]. : HT, n c(r n ) = R[ɛ] R n Thom ) µ (n) Ω n (R n ) T = U(1) µ (n) T ξ t dµ (n) = 0, L ξ µ (n) = 0. 17

18 dι ξ µ (n) = (L ξ ι ξ d)µ (n) = 0. Hc n 1 (R n ) = 0 (n 2)-fom µ (n 2) ι ξ µ (n) = dµ (n 2) T = U(1) µ (n 2) T µ (n) + ɛµ (n 2) + ɛ 2 µ (n 4) T = U(1) Lie t ɛ : t R nomalization t = e i ɛ t T = U(1) C T = U(1) R 2 = C z t k z (k Z) t ξ = ɛ z = x + i y ξ = ( ɛ = k y x + x ) y µ (2) = e 1 2 (x2 +y 2) dxdy dµ (2) = 0, ι ξ µ (2) = k e 1 2 (x2 +y 2) ( ydy xdx) ( ) = d k e 1 2 (x2 +y 2 ). µ = µ (2) + kɛ e 1 2 (x2 +y 2 ) µ = R 2 e 1 2 (x2 +y 2) dxdy = 2π, R 2 µ (0, 0) = kɛ. H 2 T, c (R2 ) = R R 2 µ µ eff = 2π kɛ µ Ω c(r 2 ) R[ɛ, ɛ 1 ] Gauss R 2 µ = µ eff (0, 0). 18

19 2.4.3 T = U(1) 2m M p M T M p M T T = U(1) (k 1 (p),..., k m (p)) Z m M µ µ = ( ) m 2π µ p ɛ k 1 (p) k m (p). T = U(1) 2m M p M T T (k 1 (p),..., k m (p)) (Z ) m M µ µ = µ p i (k i(p) ɛ), ɛ = (ɛ 1,..., ɛ ). M p M T (2π) m CP 1 = {[z 0 : z 1 ]} T = U(1) t [z 0 : z 1 ] = [z 0 : tz 1 ] [1 : 0] t [0 : 1] t 1 ( 1 0 = 1 = 2π CP ɛ + 1 ). ɛ 1 CP 2 = {[z 0 : z 1 : z 2 ]} T = U(1) 2 (t 1, t 2 ) [z 0 : z 1 : z 2 ] = [z 0 : t 1 z 1 : t 2 z 2 ] [1 : 0 : 0] t 1 + t 2 [0 : 1 : 0] t1 1 + t 1 1 t 2 [0 : 0 : 1] t t 1 t 1 0 = CP 2 1 = (2π) 2 ( 1 ɛ 1 ɛ ɛ 1 (ɛ 2 ɛ 1 ) + 1 ɛ 2 (ɛ 1 ɛ 2 ) ) M T X 2m X X N X T Eule e X e X p X 0 Thom fom, Eule fom Thom fom Thom Eule fom 19

20 Lie DG Lie Lie g g = g 0, ξ L ξ g = g 1, ξ ι ξ g 0 g 1 DG Lie d(ι ξ ) = L ξ, d(l ξ ) = 0, [L ξ, L η ] = L [ξ, η], [L ξ, ι η ] = ι [ξ, η], [ι ξ, ι η ] = 0. Lie g M Lie DG Lie g 0 g 1 DG Ω (M) DG Lie g 0 g 1 DG g-dg U(1) Weil T = U(1) gaded algeba Λt St = Λ R (θ) R[ɛ], deg θ = 1, deg ɛ = 2 d ξ t, ɛ(ξ) = 1 dθ = ɛ, dɛ = 0, ι ξ θ = 1, ι ξ ɛ = 0 T = U(1) Weil L ξ θ = 0, L ξ ɛ = 0 H ((Λt St ) basic ) = St. T = U(1) π : P B θ P : t Ω 1 (P ) t-dg Λt St Ω (P ) Weil ɛ basic B 20

21 2.5.3 The U(1) Weil model T = U(1) M ((Λt St Ω (M)) basic, d) U(1) Weil model H T (M) Weil Weil model U(1) de Rham T = U(1) S 2k+1 CP k Weil H T (M) Weil = lim k H (M T (k), R) = H T (M, R) Boel U(1) Weil model U(1) Catan model H T (M) Weil = H T (M) Catan. ) µ = k ɛ k (µ k + θµ k) (Λt St Ω (M)) basic L ξ µ = 0, ι ξ µ = 0 L ξ µ k = 0, L ξ µ k = 0, ι ξ µ k + µ k = 0, ι ξ µ k = 0. µ = k ɛ k µ k µ St Ω (M) t, µ = µ θι ξ µ. µ St Ω (M) t L ξ (µ θι ξ µ ) = 0, ι ξ (µ θι ξ µ ) = 0. d T = d ɛ ι ξ d(µ θι ξ µ ) = dµ ɛ ι ξ µ θι ξ dµ = d T µ θι ξ d T µ. 21

22 3 Thhe 3.1 Lie the femion side Lie Lie G G G G H (G) H (G) H (G) H (G). R H (G, R) = Λ R (ψ 1,..., ψ ), deg(ψ i ): odd. G deg(ψ 1 ) deg(ψ ), deg(ψ i ) = 2m i + 1 (m 1,..., m ) G exponent H (U(1), R) = H ((S 1 ), R) = Λ R (ψ 1,..., ψ ), deg(ψ i ) = 1, H (SU(2), R) = H (S 3, R) = Λ R (ψ), deg(ψ) = 3, H (U(n), R) = Λ R (ψ 1,..., ψ n ), deg(ψ i ) = 2i G G X G X X H (X) H (G) H (G) H (X) H (X) G (G-space) H (G) (H (G)-mod) H : (G-space) (H (G)-mod) 3.2 the boson side G G EG BG G (univesal pincipal G-bundle), BG (classifying space) ΩBG G. Hausdoff Y G P f : Y BG ωy = f 1 EG = y Y EG f(y) 22

23 P P EG f Y f BG BG thhe Dinfeld [D04] Lie G U(n) EU(n) G G EU(n) EU(n)/G G C n+k Hemite u, v = u v V n (C n+k ) = {(v 1,..., v n ) v i C n+k, v i, v j = δ ij }, G n (C n+k ) = {C n+k n } V n (C n+k ) Stiefel G n (C n+k ) Gassmann V n (C n+k ) G n (C n+k ), (v 1,..., v n ) span{v 1,..., v n } U(n) n = 1 V 1 (C k+1 ) = S 2k+1 G 1 (C k+1 ) = CP k ( ) S 2k+1 CP k U(1) V n (C n+k ) 2k 2k S 2k+1 V n (C n+k ) V n 1 (C n+k ), (v 1,..., v n ) (v 1,..., v n 1 ) S 2k+1 S 2k+3 S 2k+1 V n (C n+k ) V n 1 (C n+k ) S 2k+3 V n (C n+k+1 ) V n 1 (C n+k+1 ) 23

24 V n (C n+k ) V n (C n+k+1 ) U(n) V n (C n ) G n (C n ) V n (C n+1 ) G n (C n+1 ) V n (C n+2 ) G n (C n+2 ) EU(n) = V C n = k V n (C n+k ), BU(n) = G C n = k G n (C n+k ) EU(n) BU(n) U(n) EU(n) n = 1 BU(1) = CP G A (Milno Stasheff) Lie G BG H (BG, R) = R[x 1,..., x ], deg(x i ) = deg(ψ i ) + 1: even. G H (B(U(1) ), R) = H ((CP ), R) = R[x 1,..., x ], deg(x i ) = 2, 3.3 Boel H (BSU(2), R) = H (HP, R) = R[x], deg(x) = 4, f : Y BG G H (BU(n), R) = H (G C n, R) = R[c 1,..., c n ], deg(c i ) = 2i. ωy = f 1 EG = {(y, p) Y EG f(y) = π(p)} ωy EG π Y f ω G (Space/BG) G (G-space) BG 24

25 G X bx = X G = EG G X bx BG EG X EG bx BG b (G-space) (Space/BG) Boel x 0 X G EG EG X, p (p, x 0 ) bx BG b ω (G-space)(ωY, X) = (Space/BG)(Y, bx) ) ϕ (G-space)(ωY, X) ωy EG X, (y, p) (p, ϕ(y, p)) G ψ (Space/BG)(Y, bx) ψ(y) = [(p, ϕ(y, p))] ψ (Space/BG)(Y, bx) (y, p) ωy ψ(y) EG π(p) G X G EG π(p) 1 x X ψ(y) = [(p, x)] ϕ (G-space)(ωY, X) ψ(y) = [(p, ϕ(y, p))] ωbx = EG X X bωy = EG G ωy Y Y BG H (BG) H (Y ) H : (Space/BG) (H (BG)-mod) 25

26 3.3.6 G X H G(X) = H (bx) = H (X G ) H G(pt) = H (BG) H G = H b : (G-space) (H (BG)-mod) G G E X E G X G H (X G ) = H G (X) E O(1) = S 0 = Z 2 O(1) O(1) mod 2 Eule Stiefel-Whitney U(1) = S 1 U(1) U(1) Eule Chen Steenod X Z 2 X X X α H q (X, Z 2 ) α α H 2q (X X, Z 2 ) P (α) H 2q Z 2 (X X, Z 2 ) Sq(α) = q x q i Sq i (α) H (RP, Z 2 ) H (X, Z 2 ) = Z 2 [x] H (X, Z 2 ) i=0 Sq i : H q (X, Z 2 ) H q+i (X, Z 2 ) Steenod 26

27 A A.1 A.1.1 X 0 X 1 X 2 X k X k+1 X = k X k U X k U X k X k F X k F X k X k {X k } lim X k f : X Y f X k X k X U k X k X k X U X U k = U X k ) X U U X k X k U k X k X k X k+1 X k+1 U k+1 U k = U k+1 X k. X k+1 X k+2 X k+2 U k+2 U k+1 = U k+2 X k+1. i k X i U i U i = U i+1 X i U = i k U i i k U X i = U i. U X U k = U X k A.1.2 T 1 X k T 1 1 X T 1 X T 1 X k T 1 X K k K X k. ) k x k K X k U k = X {x k, x k+1, x k+2,... } U k U k+1 X k U k k U k = X K U k 27

28 i X i {x k, x k+1, x k+2,... } X i T 1 U k X i = X i (X i {x k, x k+1, x k+2,... }) X i U k X K X q σ : q X k σ( q ) X k q = {(t 1,..., t q ) 0 t 1 t q 1}. A.2 A.2.1 (X, A) A X π n (A) π n (X) π n (A) g : (I n, I n ) (A, ) π n (X) h : I n I X h(i n {0}) = h( I n I) =, h(, 1) = g π n+1 (X, A) h : I n I X h(i n {0}) = h( I n I) =, h(i n {1}) A. h h(, 1) π n+1 (X, A) π n (A) π n+1 (X, A) π n (A) π n (X) π n+1 (X, A) h π n (A) g 1 : I n I A g 1 (I n {0}) = g 1 ( I n I) =, g 1 I n {1} = h I n {1}. h g 1 π n+1 (X) π n+1 (X) π n+1 (X, A) π n (A) π n+1 (X) f : (I n+1, I n+1 ) (X, ) π n+1 (X, A) f 1 : I n I I X f 1 (I n I {0}) = f 1 (I n {0} I) = f 1 ( I n I I) =, f 1 (,, 1) = f, f 1 ( (I n I) I) A. 28

29 f 1 f f 1 (, 1, ) : I n+1 A π n+1 (A) π n+1 (X) π n+1 (X, A) π n+1 (A) π n+1 (X) π n+1 (X, A) π n (A) π n (X). A.2.2 π : E B S (coveing homotopy popety: CHP) (homotopy lifting popety: HLP) F : S I B f : S E F (, 0) = π f S ef E π S I F B F : S I E F = π F π : E B See fibation I n (n = 0, 1, 2,... ) CHP π : E B, π( E ) = B quasi-fibation F = π 1 ( B ) π n (E, F ) π n (B) n 1 A quasi-fibation π : E B π n+1 (F ) π n+1 (E) π n+1 (B) π n (F ) π n (E) See fibation quasi-fibation. A.3 A.3.1 (C, d) exhaustive deceasing filtation C n = F 0 C n F 1 C n F 2 C n, F p C n = 0 p d(f p C n ) F p C n+1 29

30 (F C, d) filteed complex C 2 d 1 filtation K p, q = {x F p C p+q dx F p+ C p+q+1 } L p, q = {dy F p C p+q y F p +1 C p+q 1 } filtation filtation g d g A.3.2 filteed complex (F C, d) K p, q = {x F p C p+q dx F p+ C p+q+1 } = F p C p+q d 1 (F p+ C p+q+1 ), L p, q = {dx F p C p+q x F p +1 C p+q 1 } = df p +1 C p+q 1 F p C p+q K p, q dk p, q p+, q +1 = L+1. K p, q +1, Lp, q +1 Lp, q K p, q L p, q := := K p, q = {x F p C p+q dx = 0}, L p, q = L p, q p+1 = {dx F p C p+q x C p+q 1 } dd = 0 K p, q L p, q F p C = K p 0 Kp 1 Kp L p L p 1 Lp 0 = df p+1 C. 30

31 F p C F p+1 C K p F p+1 C = K p+1 1, Kp F p+1 C = K p+1, L p F p+1 C = L p+1 +1, Lp F p+1 C = L p+1. A.3.3 filtation filtation A p, q B p, q = K p, q + F p+1 C p+q, A p, q = K p, q + F p+1 C p+q = = L p, q + F p+1 C p+q, B p, q = L p, q + F p+1 C p+q = A p, q, B p, q F p C F p+1 C filtation F (p) : F p C = A p 0 Ap 1 Ap B p B p 1 Bp 0 = F p+1 C p C filtation (F (p)) p 0 g (F (p)) p 0..., B p 1 1 B p 1 0, A p 0 A p, 1 A p 1 A p,..., 2 A p B p,..., B p 2 B p, 1 B p 1 B p, 0 A p+1 0 A p+1 1,..., d A.3.4 dk p = L p+ +1, dkp +1 = Lp++1 +2, dk p+1 1 = Lp+, K p F p+1 C = K p+1 1, d g F (p) A p /A p +1 Lp+ +1 F p++1 C = L p g F (p + ) Bp+ +1 /Bp+ A p A p +1 K p = K p +1 + Kp+1 d 1 L p+ L p L p = B p+ +1 B p+ K p +1 + Kp+1 1 Kp +1 Kp = K p Ke d (g F (p)) p 0 (A p /B p ) p 0 d 31

32 A.3.5 F p C C HC F p HC HC exhaustive deceasing filtation HC = F 0 HC F 1 HC F 2 HC, F p HC = 0 p g F (p) A p /B p gp HC A p B p = Kp + F p+1 C L p + F p+1 C = K p = L p + K p+1 = F p HC/F p+1 HC = g p HC. {x F p C dx = 0} {dy F p C} + {x F p+1 C dx = 0} A.3.6 A p A p +1 Bp +1 Bp A p+ A p+ +1 Bp+ +1 Bp+ A p B p B p +1 Ap +1 suj. A p A p = +1 d p : Ap B p B p+ +1 B p+ Ap+ B p+ A p+ inj. B p+ B p +1 B p Ap B p Ap A p +1 0 d d p d p = 0. d : K p K p+ A p B p K p = L p + K p+1 d 1 L p+ K p+ + K p++1 1 dd = 0 = A p+ B p+. E p, q = Ap, q B p, q, E p, q = Ap, q B p, q 32

33 d p, q (E, d ) : E p, q ( A Ke d p p = Ke Im d p = Im Ke d p Im d p B p (E, d ) p+, q +1 E. Ap A p +1 ( B p +1 B p Ap B p ) ) = Ap +1 B p = E p = Ap +1 B p, = Bp +1 B p A p A p +1 = B p+ +1 B p+ 1 E +1 = 0, 1 K p 0 = F p C, L p 0 = df p+1 C F p+1 C E p 0 = F p C/F p+1 C = g p C. E p 1 = H(g p C). (g F (p)) p 0 s d s = 0 E p, q = E p, q +1 = = Ep, q = g p H p+q C. A.3.7 (E p, q, d p, q : E p, q p+, q +1 E, E p, q (spectal sequence) +1 = Ke d p, q p, q+ 1 /Im d ) 33

34 A.3.8 Gysin filteed complex F C E p, q 2 = 0 fo q 0, n E p, q 2 = E p, q 3 = = E p, q n+1, Ep, q n+2 = Ep, q n+3 = = Ep, q, 0 E p, q n n+2 Ep, n+1 E p+n+1, 0 n+1 E p+n+1, 0 n+2 0 exact, d n+1 0 E p, 0 H p C E p n, n 0 exact. E p n 1, n 2 E p, 0 2 H p C E p n, n 2 E p+1, 0 2 H p+1 C Gysin A.3.9 C, C m = p Cp, m p filtation F p C = C k,, k p 2 F q C = l q C, l 1. 0 A B C 0 i j filtation 2. filtation E filtation (a) E (b) E 1 d 1 0 HA i HB j HC 0 0 (c) E 2 Ke i, Ke j /Im i, Coke j d 2 Ke i Coke j 0 (d) E 3 0 Im i = Ke j d 2 : Ke i Coke j 34

35 5. H p A H p B H p C H p+1 A H p+1 B H p+1 C H p C H p+1 A d 2 1 (five lemma) f 1 f 2 f 3 f 4 A 1 A2 A3 A4 A5 g 1 g 2 g 3 g 4 g 5 A 1 f 1 A 2 f 2 A 3 f 3 A 4 f 4 A 5 g 2, g 4 g 1 g 5 g 3 1. A i i A i i + 1 filtation 2. filtation E 1 Ke f Coke f 4 Ke f Coke f 4 3, , 4 4. filtation (a) E 3, 4 (b) E 1 Ke g 1 0 Ke g E (c) g Coke g 3 0 Coke g 5 35

36 Refeences [At85] M. F. Atiyah, Cicula symmety and stationay-phase appoximation, Astéisque 131(1985), [AB84] M. F. Atiyah and R. Bott, The moment map and equivaiant cohomology, Topology 23(1984), 1 28 [BT82] R. Bott and L. W. Tu, Diffeential Foms in Algebaic Topology, Spinge GTM 82, 1982 [D04] V. Dinfeld, DG quotients of DG categoies, J. Alg. 272(2004), math.kt/ [GS99] V. W. Guillemin and S. Stenbeg, Supesymmety and Equivaiant de Rham Theoy, Spinge, 1999 [Ha02] A. Hatche, Algebaic Topology, Cambidge Univ. Pess, 2002, [Hu75] D. Husemolle, Fibe Bundles, 2nd ed., Spinge GTM 20, 1975 [ 77]

E1 (4/12)., ( )., 3,4 ( ). ( ) Allen Hatcher, Vector bundle and K-theory ( HP ) 1

E1 (4/12)., ( )., 3,4 ( ). ( ) Allen Hatcher, Vector bundle and K-theory ( HP ) 1 (4/12) 1 1.. 2. F R C H P n F E n := {((x 0,..., x n ), [v 0 : : v n ]) F n+1 P n F n x i v i = 0 }. i=0 E n P n F P n