topology.dvi

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1 LastUpdate: Kunneth Tor Ext Kunneth Mayers-Vietoris CW Cech ( ) ( ) ( ) ( ) ( ) Euler Lefschetz RP n

2 Lie Lie Manifolds h Poincare Rokhlin Donaldson K Chern Clifford Dirac R n Atiyah-Singer

3 Poincaré-Hopf Euler Thom Z 2 -Euler Thom Stiefel-Whitney Chern Pontrjagin Knots and Links (linking number) (bridge index) (braid index) (unknoting number) Seifert (genus) (signature) Alexander-Conway Skein Jones Skein State Homfly Skein Q Skein Kauffman

4 Skein Yang-Baxter

5 1 [LastUpdate: ] ( ) R C =(C j ) =( j ), j+2 Cj+1 j+1 Cj j Cj 1 j 1 2 =0 C =(C j, ) R φ : C D φ = φ φ 1.2 ( ) C,D f,g : C D R Φ n : C n C n+1 f g = n+1 Φ n +Φ n 1 n Φ f g f = g : H (C ) H (D ) 1.3 ( ) C H (C )=Z (C )/B (C ); Z (C )=Ker, B (C )=Im R H (C )=(H j (C )),Z (C )=(Z j (C )),B (C )= (B j (C )) H (C ) C Z B φ : C D R φ : H (C ) H (D ) φ ([c]) = [φ(c)] R R H 5

6 1.4 ( ) 1) 3 (A, ), (X, ), (Y, ) 0 A f X y Z n (Y ) g Y 0 [y] =[f 1 g 1 (y)] R- : H n (Y ) H n 1 (A) Hn (A) f Hn (X) 2) g Hn (Y ) 0 A X Y 0 φ ψ 0 A X Y 0 Hn 1 (A) ψ = φ : H (Y ) H (A) f ( ) R C =(C j ) δ =(δ j ), δ j 2 C j 1 δ j 1 C j δ j C j+1 δ j+1 δ 2 =0 C =(C j,δ) R φ : C D δφ = φδ φ 6

7 1.6 ( ) C H (C )=Z (C )/B (C ); Z (C )=Kerδ, B (C )=Imδ R H (C )=(H j (C )),Z (C )=(Z j (C )),B (C )= (B j (C )) H (C ) C Z B φ : C D φ [u] =[φ(u)] R φ : H (C ) H (D ) R R H 1.7 ( ) 1) 3 (Y,δ ), (X, δ), (A, δ ) 0 Y f X g A 0 a Z n (A) δ [a] =[f 1 δ g 1 (a)] R- δ : H n (A) H n+1 (Y ) δ H n (Y ) f H n (X) 2) g H n (A) 0 Y X A 0 φ ψ 0 Y X A 0 δ H n+1 (Y ) δ ψ = φ δ : H (A) H (Y ) f 7

8 1.2 Kunneth Tor Ext 1.8 ( ) R 1) R- A R- F 0,F 1 R d : F 1 F 0,ɛ: F 0 A 0 F 1 d F 0 ɛ A 0 A 2) R A, B A 1) 0 F 1 R B d 1 F 0 R B 0 A B Tor R (A, B) Tor R (A, B) =kerd 1 0 Tor R (A, B) F 1 R B F 0 R B A R B ( ) i) C R Tor C C C ii) Tor(A, B) =Tor(B, A). iii) Tor(A B,C) =Tor(A, C) Tor(B,C). iv) A Tor Z (A, B) =0. v) K 0 A Tor Z (K, A) =0. 8

9 vi) Tor Z (Z m, Z n ) = Z d (m, n > 0, d =GCD(m, n)). vii) K Tor K (A, B) = (Ext) A R A, B R 0 F 1 d F 0 0 Hom R (F 0,B) ɛ A 0 d T Hom R (F 1,B) 0 Ext R (A, B) Ext R (A, B) =Hom R (F 1,B)/Im d T. 0 Hom R (A, B) Hom R (F 0,B) Hom R (F 1,B) Ext R (A, B). Ext R (A, B) A 1.11 (Ext ) Ext i) C R Ext C C C ii) R A, B, C iii) A Ext R (A B,C) = Ext R (A, C) Ext R (B,C), Ext R (A, B C) = Ext R (A, B) Ext R (A, C). Ext Z (Z,B)=0, Ext Z (Z m,b) = B/mB (m >0). 9

10 ( ( )) R C R G R 0 H n (C ) G H n (C G) Tor R (H n 1 (C ),G) ( ( )) R C R G R 0 Ext R (H n 1 (C ),G) H n (Hom R (C,G)) Hom R (H n (C ),G) Kunneth 1.14 (Künneth ( )) R C R D R 0 p+q=n H p(c ) R H q (D ) H n (C R D ) p+q=n 1 TorR (H p (C ),H q (D )) 0 D R 1.15 (Künneth ( )) R C R D R 0 p+q=n Hp (C ) R H q (D ) H n (C R D ) p+q=n+1 TorR (H p (C ),H q (D )) 0 D R 10

11 ( ) 1) K Σ (K, Σ) i) s Σ s s(s ) s Σ. ii) v K {v} Σ Σ. s Σ K (n +1) s n( ) 0 K 2) (K, Σ), (K 0, Σ 0 ) K 0 K Σ 0 Σ K 0 K 3) K, L φ : K L K s = {v 0,,v n } {φ(v 0 ),,φ(v n )} L φ K, L φ φ φ 1 K L 3) K q s s s q q σ s {v 0,,v q } σ =[v 0,,v q ] 1.17 ( ) 11

12 1) K q Abel C q (K) q : C q (K) C q 1 (K) q q ([v 0,,v q ]) = ( 1) k [v 0,,v k 1,v k+1,,v q ] k=0 σ q σ q C =(C q, q ) H q (K) K R G R C Z G H (K; G) K G R C =Hom Z (C,G) H (K; G) G 2) X K H (K; G),H (K; G) K H (K; G) H (K; G) X ( ) 1) X R n+1 Δ n X σ :Δ n X X n- S n (X) S (X) = (S n (X)) n <0 S n (X) =0 n : S n (X) S n 1 (X) n n (σ) = ( 1) j σ ɛ j, σ S n (X) j=0 12

13 ɛ j Δ n 1 e k (k =0,n 1) Δ n e k (k j) e k+1 (k>j) Δ n 1 Δ n S (X) =(S j (X), ) 2) (X, A) A X S (X) S (A) S (X, A) :=S (X)/S (A) Z- S R G C C G Hom Z (C,G) R- G R- Hom(,G) H G S G (X, A) H (X, A; G) (X, A) G H Hom(,G) S G (X, A) H (X, A; G) (X, A) G H (X, A; G) =H ((S (X)/S (A)) G), H (X, A; G) =H (Hom(S (X)/S (A),G)) S (X, A; G) =(S (X)/S (A)) G, S (X, A; G) =Hom(S (X)/S (A),G) R = G = Z G c S n (X) [c] Z n (X, A) c S n (A) [c] B n (X, A) c S n+1 (X) s.t. c c S n (A) 13

14 2. R 0 A B C Hom(C, G) Hom(B,G) Hom(A, G) 0 S n (X, A; G) u Hom(S n (X),G) u Sn(A) =0 < δ[u], [c] >=< u, c> [u] Z n (X, A; G) <u, c>=0 c S n+1 (X) [u] B n (X, A; G) [u ] Hom(S n 1 (X),G) s.t. <u,c>=< u, c> c S n 1 (X) (( ) ) G (X, A) 1) 0 H n (X, A; Z) G H n (X, A; G) Tor Z (H n 1 (X, A; Z),G) 0 2) 0 Ext Z (H n 1 (X, A; Z),G) H n (X, A; G) Hom(H n (X, A; Z),G) ( ) 14

15 1. (X, A) H n+1 (X, A; G) j Hn (X, A; G) Hn (A; G). A H n+1 (X, A; G) j Hn (X, A; G) Hn (A; G). i Hn (X; G) i Hn (X; G) 2. B A X H n+1 (X, A; G) j Hn (X, A; G) Hn (A, B; G). i Hn (X, B; G) 1.22 ( ) 1. (X, A) H n 1 (A; G) i H n (A; G) δ H n (X, A; G) δ. A H n 1 (A; G) i H n (A; G) δ H n (X, A; G) δ. j H n (X; G) j H n (X; G) 15

16 2. B A X H n 1 (A, B; G) δ H n (X, A; G) j H n (X, B; G) i H n (A, B; G) δ Mayers-Vietoris 1.23 ( ) X X 1,X 2 S(X 1 )+S(X 2 ) S(X 1 X 2 ) {X 1,X 2 } Y = X 1 X 2 X i int Y X i Y =int Y X 1 int Y X 2 {X 1,X 2 } 1.24 (Mayers-Vietoris ) X {X 1,X 2 } X Δ H n (X 1 X 2 ; G) i 1 ( i 2 ) H n (X 1 ; G) H n (X 2 ; G) j 1 +j 2 Hn (X 1 X 2 ; G) Δ H n (X 1 X 2 ; G) i 1 i 2 H n (X 1 X 2 ; G) Δ H n 1 (X 1 X 2 ; G), j 1 (j 2 ) H n (X 1 ; G) H n (X 2 ; G) Δ H n 1 (X 1 X 2 ; G) X 1 X 2 H q (H q ) H q ( H q ) 16

17 1.25 ( ) U A X X V VA X V X U r t r t (A V ) A V i :(X U, A U) (X, A) G i : H n (X U, A U; G) = i : H n (X, A; G) = H n (X, A; G), H n (X U, A U; G) CW 1.26 (CW ) 1) Hausdorff e φ :(D n,s n 1 ) (ē, ė) e n( ) φ 2) Hausdorff X {e λ λ Λ} X i) e μ e ν = (μ ν). ii) X = λ Λ e λ. iii) X q := μ:dim eµ qe μ dim e μ = q+1 ė μ X q. 3) Hausdorff X X q q X A X e ē A A A A X X x X x inta A X 4) X Y f f(x q ) Y q 17

18 5) X CW C) X e e W) X U X e U ē ē 1.27 (CW ) CW 1) CW 2) CW X A CW 3) CW 4) ( ) CW 5) CW 6) CW 7) CW X, Y X Y X Y CW 8) CW X, Y X Y CW 9) K (ē q = D q ) CW CW 18

19 1.28 CW X A (X, A) X q = X q A G H q ( X n, X n 1 ; G) =0(q n), φ λ : H n (Dλ n,sn 1 λ ; G) = H n ( X n, X n 1 ; G) λ Λ n λ Λ n H n ( X n, X n 1 ) 1.29 (CW ) 1) C n (X, A) C n (X, A) :=H n ( X n, X n 1 ) : C n (X, A) C n 1 (X, A) ( X n, X n 1 ) j : H n ( X n, X n 1 ) H n 1 ( X n 1 ) j H n 1 ( X n 1, X n 2 ) (C (X, A), ) CW (X, A) 2) CW (X, A) C (X, A) H (C (X, A) G) H (Hom(C (X, A),G) CW Cech ( ) 1.30 (Čech ) X U U n n U K(U ) X A (K(U ),L(U )) H (K(U ),L(U ); G) 19

20 H (K(U ),L(U ); G) X (X, A) Čech Ȟ (X, A; G) = lim H (K(U ),L(U ); G), Ȟ (X, A; G) = lim H (K(U ),L(U ); G) ( ) 1.31 (Eilenberg-Steenrod ) 1) H i) (X, A) q Abel H q (X, A) ii) f :(X, A) (Y,B) q f : H q (Y,B) H q (X, A) iii) (X, A) q δ : H q H q+1 (X, A) H H I) 1 =1 II) f :(X, A) (Y,B) g :(Y,B) (Z, C) (g f) = f g III) ( ) f g :(X, A) (Y,B) f = g 20

21 IV) ( δ H q (X, A) δ H q+1 (X, A) j H q (X) j. i : A X j :(X, φ) (X, A) i H q (A) V) f :(X, A) (Y,B) f δ = δ (f A ) VI) ( X U Ū inta i : H q (X, A) = H q (X U, A U). VII) ( H q (pt) = 0 (q 0) Hausdorff 2) CW VI) X 1,X 2 CW i : H q (X 1 X 2,X 2 ) = H q (X 1,X 1 X 2 ) ) CW Čeck CW 2) G 3) G Čeck Čeck Hausdorff 21

22 4) G Čeck CW ( ( ) 1.33 ( ) 1) X X x y Ω(X; x, y) M(y, x) X M(y, x) X Π(X) 2) X Π(X) C S C X 3) C S γ M(x, x) S (γ) Aut(S x ) S x : π 1 (X, x) Aut(S x ) S x ) X x 0 X, C C X π 1 (X, x 0 ) Aut(A)(A C ) 2) X X C S S x 22

23 1.35 ( ) X S S (γ) :S (y) S (x) γ M(y, x) 1.36 (n ) X {π n (X, x); x X} X n X S R X X S q Šq(X; S ) Š(X; S )=(Šq(X; S ), ) X Ȟ (X; S ) X X S {K} {S (X, X K; S )} Š (X; S ) := lim K S (X, X K; S ) Ȟ (X; S ) 1.39 X S Ȟ (X; S ) = lim K H (X, X K; S ) 23

24 Ȟ (X; S ) H (X; S ) X 1.4 ( ) ( ) 1.40 ( ) X X 1,X 2 S(X 1 )+S(X 2 ) S(X 1 X 2 ) (X 1,X 2 ) 1.41 ( ) 1) X Y n (σ, τ) ρ(σ, τ)= n i+1 n σ 0 i 1 τ i=0 ρ : S(X Y ) S(X) S(Y ) Alexander-Whitney ρ κ : S(X) S(Y ) S(X Y ) 2) (X, A), (Y,B) : H p (X, A; G 1 ) H q (Y,B; G 2 ) H p+q (S(X, A; G 1 ) S(Y,B; G 2 )) κ Hp+q (S((X, A) (Y,B); G 1 G 2 )) {A Y,X B} (X, A), (Y,B) CW κ 24

25 3) (X, A), (Y,B) {A Y,X B} (X, A), (Y,B) CW q : H p (X, A; G 1 ) H q (Y,B; G 2 ) H p+q (S (X, A; G 1 ) S (Y,B; G 2 )) ρ H p+q (S ((X, A) (Y,B); G 1 G 2 )) 1.42 (( ) Künneth ) R 1) (X, A), (Y,B) {A Y,X B} 0 p+q=n H p(x, A; R) R H q (Y,B; R) H n ((X, A) (Y,B); R) p+q=n 1 TorR (H p (X, A; R),H q (Y,B; R)) 0 2) (X, A), (Y,B) CW X CW 0 p+q=n Hp (X, A) Z H q (Y,B) H n ((X, A) (Y,B)) p+q=n+1 TorZ (H p (X, A),H q (Y,B)) ( ) X A 1,A 2 {A 1 X, X A 2 } X X {A 1,A 2 } X Δ:X X X : H p (X, A 1 ; G 1 ) H q (X, A 2 ; G 2 ) H p+q ((X, A 1 ) (X, A 2 ); G 1 G 2 ) Δ H p+q (X, A 1 A 2 ; G 1 G 2 ) 25

26 u S p (X; G 1 ),u 2 S q (X; G 2 ) σ S p+q (X) σ, u v := p+1 p+q σ, u 0 p 1 σ, v G 1 G 2 δ(u v) =(δu) v +( 1) p u (δv), [u] [v] =[u v] 1.44 ( ) u H p (X, A 1 ; R),v H q (X, A 2 ; R),w H r (X, A 3 ; R) u v 1) f : Y X f(b i ) A i (i =1, 2) f u f v f (u v) =f (u) f (v) H p+q (Y,B 1 B 2 ; R) 2) ( ) (u v) w = u (v w) 3) R 1 H 0 (X; R) u 1=1 u = u H 0 (X, A 1 ; R) 4) R v u =( 1) pq u v H p+q (X, A 1 A 2 ; R) 1.45 ( ) G 1,G 2 c = i σ i g i S n (X; G 1 ),u S p (X; G 2 ) c u S n p (X; G 1 G 2 ) c u = i 0 p 1 σ i (g i p+1 n σ i,u ) 26

27 (c u) =( 1) p ( c u c δu) A, B X : H n (X, A B; G 1 ) H p (X, A; G 2 ) H n p (X, B; G 1 G 2 ) [c] [u] =[c u] 1.46 ( ) R X, Y A, A i X B i Y 1) {A 1,A 2 } X {B 1,B 2 } Y f : X Y f(a i ) B i (i =1, 2) c H n (X, A 1 A 2 ; R),v H p (Y,B 1 ; R) f (c f v)=f (c) v H n p (Y,B 2 ; R) 2) c H n (X, A 1 A 2 A 3 ; R),u H p (X, A 1 ; R),v H q (X, A 2 ; R) c u (c u) v = c (u v) H n p q (X, A 3 ; R) 3) 1 H 0 (X; R) c H n (X, A; R) c 1=c. 4) c H n (X, A; R),u H n (X, A; R) ɛ (c u) = c, u R ɛ ɛ : S 0 (X) Z H 0 (X; R) R 27

28 Euler Lefschetz 1.47 ( ) (X, A) n H n (X, A) =0 (X, A) H n (X, A) T n H n (X, A) =H n (X, A)/T n f :(X, A) (X, A) f (n) : Hn (X, A) H n (X, A) L(f) := n=0 ( 1) n trf (0) f Lefschetz L(1) χ(x, A) := ( 1) n b n n=0 (X, A) Euler 1.48 (Hopf) (X, A) CW f :(X, A) (X, A) C n (X, A) f (n) n L(f) = n=0 trf (0) f =id X n A c n χ(x, A) = n ( 1) n c n ( ) X {X 1,X 2 } X 1,X 2 A = X 1 X 2 X = X 1 X 2 f : X X f(x i ) X i L(f)+L(f A) =L(f X 1 )+L(f X 2 ) 28

29 χ(x)+χ(a) =χ(x 1 )+χ(x 2 ) ( ) X, Y X Y f : X X, g : Y Y L(f g) =L(f)L(g) χ(x Y )=χ(x)χ(y ) X H q (X I,X I) = H q 1 (X), H q (X I,X I) = H q 1 (X) ( ) X, Y n n 3 H q (X Y ) = H q (X) H q (Y ); 1 q n 2 X, Y H n 1 (X Y ) = H n 1 (X) H n 1 (Y ) 29

30 1.53 (Euler ) X, Y n χ(x)+χ(y ) 1 ( 1) n ; X, Y compact, χ(x Y )= χ(x)+χ(y ) 1+( 1) n ; X, Y noncompact, χ(x)+χ(y ) 1 ; the other cases RP n 1 S n CW CW D k R k u k :Δ k =(0, 1,,k) (v 1,,v k, 0) D k H k (D k,d k (v v k )/k) = H k (D k,d k 0) v j j R n 1 R n S n R n+1 k ē k ± = S n R k ± (k =0, 1,,n) S n = e 0 + e0 en + en. j : R k+1 R k j(x 1,,x k+1 )=(x 1,,x k ) e k ± j [ē k ±, ēk ± ]=±[Dk, D k ] D± k ± (v 1,,v k 1, ±v k, 0) ±( 1) k 1 (v 1,,v k 1, 0) ± ( 1) k (v 1,,v k 1, ±v k )+ j (v 1,,v k 1, ±v k )=(v 1,,v k 1, 0) D k 1 30

31 [D k ±, D k ±]= ( 1) k D k 1 +( 1) k e k 1 ± e k ± = ± [Dk, D k 1 ]=±( 1) k (e k ek 1 2) T : R n+1 R n+1 T (x) = x π : S n RP n = S n / {1,T}. )(k =1,,n). RP n e k = π (e k + ) e 0,,e n RP n RP n = e 0 e 1 e n. π (e k )=( 1) k+1 e k e k =(1+( 1) k )e k 1. 0 C n (1+( 1) n ) C n 1 C 2 2 C 1 0 C 0 0. H 0 (RP n )=Z, H 2k (RP n )=0(k>0), H 2k 1 (RP n )=Z 2 (2k 1 <n), H 2k 1 (RP n )=Z (2k 1=n). 2 [LastUpdate: ] 31

32 2.1 Lie Lie 2.1 ( ) n k Lie G n k 2 π k (U) = π k (U(n)) = π k (SU(n)), n (k +1)/2, (1) π k (O) = π k (SO(n)), n k +2, (2) π k (Sp) = π k (Sp(n)), n (k 1)/4. (3) 2.2 (Bott ) { π k (U) ; k 1(mod2), = 0 ;k 0(mod2). ; k 3, 7(mod8), π k (O) = π k (O) = 2 ;k 0, 1(mod8), 0 ;k 2, 4, 5, 6(mod8). ; k 3, 7(mod8), 2 ;k 4, 5(mod8), 0 ;k 0, 1, 2, 6(mod8). (4) (5) (6) 2.3 ( ) G Lie G =SO(n)(n 2), Spin(n)(n 3), U(n)(n 1), SU(n)(n 2), Sp(n)(n 1), G 2,F 4,E 2,E 6,E 8. 1) π 1 (G): ; G =U(n)(n 1), SO(2), π 1 (G) = 2 ;G =SO(n)(n 3), 0 ; (7) 32

33 2) π 2 (G): π 2 (G) =0. (8) 3) π 3 (G): 0 ;G = U (1) = SO(2) π 3 (G) = + ; G = SO(4) Spin(4) ; (9) 4) π 4 (G): 2+2 ;G = SO(4) Spin(4) π 4 (G) = 2 ;G = Sp(n), SU(2), SO(3), SO(5), Spin(3), Spin(5) 0 ;G =SU(n) (n 3), SO(n)(n 6),G 2,F 4,E 6,E 7,E 8 (10) 5) π 5 (G): π 5 (G) = 2+2 ;G = SO(4) Spin(4), 2 ;G = Sp(n), SU(2), SO(3), SO(5), Spin(3), Spin(5) ; G =SU(n) (n 3), SO(6), Spin(6) 0 ;G =SO(n), Spin(n) (n 7),G 2,F 4,E 6,E 7,E 8 (11) 3 Manifolds [LastUpdate: ] ( ) n Euclide R n {x =(x 1,,x n ) R n x n 0} H n {x H n x n =0} H n i) Hausdorff M p H n U(p) M n 33

34 ii) n M U U H n ψ (U, ψ) S = {U α,ψ α } α A {U α } α A M iii) n M S = {U α,ψ α } α A M = α A ψα 1 ( H n ) M 4 [LastUpdate: ] Morse [Morse M. (1934)] [Cairns SS (1935), Whitehead JHC (1940)] 1944 Whitney Whitney : n C R 2n 1 R 2n [Whitney H(1944)] 1952 Rokhlin [Rokhlin (1952)] 1954 Thom [Thom R (1954)] 1960 Stiefe-Whitney Pontryagin [Wall CTC (1960)] 1961 Morse [Smale S (1961), Thom R, Wallace, Morse M] n 5 Poincare [Smale S (1961)] 1962 h- : 5 C h- [Smale S(1962)] 34

35 [Smale S (1962)] 2 C [Smale S (1962)] 1963 n 1 2n [Wall CTC (1962)] n 1 2n +1 [Tamura I(1963), Wall CTC (1963)] [Kervaire MA and Milnor JW (1963)] Atiyah-Singer [Atiyah and Singer (1963)] Poincare [Freedman MH(1982)] 1982 R 4 [Donaldson SK (1983)] 1985 Donaldson [Donaldson SK] h [Donaldson SK] ( ) M n C 1. C 1 f df p =0 p M f 2. p M C 2 f H μν (p) =( μ ν f)(p) p Hesse 3. p M C 2 f Hesse p Hesse 4. M C f M M f Morse 35

36 4.2 M n C M Morse 4.3 ( ) 1. W C W V 0,V 1 W = V 0 V 1,V 0 V 1 =. (W ; V 0,V 1 ) C 2. C (W ; V 0,V 1 ) Morse f (W ; V 0,V 1 ) Morse i) f(w )=[a, b] (a<b). ii) V 0 V 0 = f 1 (a) V 1 V 1 = f 1 (b) 4.4 C (W ; V 0,V 1 ) Morse h 4.5 (h ) W n n C (W n ; V 0,V 1 ) C W n = V 0 I V 0 = V 1 36

37 i) W n,v 0,V 1 ii) n 6. iii) H q (W n,v 0 )=0(q =0, 1, 2, ). 4.6 (h ) n V,V V V = W n+1 V W n+1 V W n+1 V V h H (W n+1,v)=0 4.7 (Smale: h ) V,V n C n 5 V V h V V C [Smale, S.: On the structure of manifolds, Amer. J. Math. 8, (1962); Smale, S.: Lectures on h-cobordism theorem,princeton Univ. Press (1965)] [ ] 4.8 (Kirby-Siebenmann: h ) V,V n n 5 V V h V V [Kirby, R.C. and Siebenmann, L.C.: Foundational Essays on Topological Manifolds, Smoothings, and Triangulations, Ann. Math. Studies 88, Princeton (1977)] Poincare 4.9 (Stallings 1960; Zeeman 1961) M n n 5 S n Euclid PL M n S n PL [Stallings J 1960[Sta60]; Zeeman CW 1961[?,?]] 37

38 4.10 (Smale 1960) W n n C W n n D n C i) n 6. ii) W n H q (W n )=H q (D n )(q =0, 1, ). iii) W n 4.11 M n S n n C n =5 6 n +1 C W n+1 W n+1 = M n 4.12 (n 5 Poincare [Smale]) M n n C S n n 5 M n S n C 0 n =5, 6 M n S n C [Smale S 1960, 1961[Sma60, Sma61]] 4.13 W 5 5 C W 5 5 D 5 C i) W 5 H q (W 5 )=H q (D 5 )(q =0, 1, ). ii) W 5 S 4 C 4.14 (Schoenflies ) f : S n 1 S n C f(s n 1 ) S n S n f(s n 1 )=A 1 A 2 n 5 M 1 = A 1 f(s n 1 ) M 2 f(s n 1 ) D n C 38

39 ( ) Q : Z m Z m Z 1) v Z m Q(v, v) 0(mod2) Q II II I 2) Q (signature) 4.16 ( ) Q 1) Q II 2) Q II Q (1) ( 1) 3) Q I Q E 8 ( ) 4) Rokhlin 4.17 ( Rokhlin ) 4 C E 8 C 39

40 ( [Milnor (1956)]) M,N 4.20 ([Wall(1964)]) M,N h M N h k M k (S 2 S 2 ) N k (S 2 S 2 ) 4.21 (Freedman [Freedman (1982)]) D 2 R 2 Casson 4.22 ( h [Freedman]) N 5 N = M + M M + M = M ± H (M ; Z) H (N; Z) N M [0, 1] [Freedman, M.H.: The topology of 4-dimensional manifolds, J. Diff. Geom. 17 (1982), ] 4.23 ( h [Freedman(1982)]) C M,N h 4.24 ( Poincare [Freedman (1982)) ] S 4 S 4 [Freedman, M.H.: The topology of 4-dimensional manifolds, J. Diff. Geom. 17 (1982), ] 4.25 ( ) (Freedman-Quinn [Freedman (1982), Quinn (1982)]) X ks(x) H 4 (X; Z 2 ) Kirby Siebenmann 40

41 i) ks(x) ii) ks(x) =0 X S 1 iii) I ks(x) =0, 1 iv) II ks(x) 1/8 [Freedman, M.H. and Quinn, F.: Topology of 4-manifolds, Princeton Math. Ser. 39, Princeton, 1990] Donaldson 4.27 (Donaldson [Donaldson(1983,1987)]) C b b = ( 1) ( 1) ( 1). b 4.28 (Donaldson(1983), Taubes(1986)) R 4 C [Kirby and Siebermann (1977), Moise(1952)] ( ) i) ( ) M 1,M 2 (M 1 M 2 = ) M 1 M 2 M 1 + M 2 ii) ( ) M M 41

42 iii) n M n,v n n +1 W n+1 W n+1 = M n +( V n ) M n V n (oriented cobordant) W n+1 W n+1 iv) M n n Ω n Ω n (n =0, 1, ) Ω = Ω n 4.30 i) Ω n {M n 1 }, {M n 2 } {M n 1 } + {M n 2 } := {M n 1 + M n 2 } Ω n { } ii) Ω {M n }, {N m } {M n } {N m } := {M n N m } Ω 4.31 (Thom) Ω n, i) n 0mod4 42

43 ii) n =4m n [ ] 4.32 Ω Q CP 2, CP 4, Q [ ] 4.33 (Wall) n Pontrjagin Stiefel-Whitney Ω n Pontrjagin Z Stiefel-Whitney Z 2 [Wall, C.T.C: Determination of the cobordism ring, Ann. of Math. 72 (1960)] 5 [LastUpdate: ] K 5.1 (K ) X F - V F (X) Grothendiek K K F (X) K F (X) F = C F = R K F (X) K(X) KO(X) 5.2 X K K F (X) i) K F (X) K F 43

44 ii) K F (X) V N θf N (N 0) [V ] [θn F ] iii) K F (X) [V ] [W ]=0 θ N F V θn F = W θ N F 5.3 ( ) X pt K F (X) K F (pt) = Z K K F (X) 5.4 ( ) X V W θ m,θ n V θ m = W θ n V W 5.5 X K K F (X) i) K F (X) K F (X) K F ii) iii) K F (X) V V N θ N F (N 0) [V ] [θn F ] K F (X) [V ]=[W ] V W K F (X) X F ( K ) X Y K F (X, Y ) K F (X, Y ):= K F (X/Y ) X/Y Y Y = K F (X, ) :=K F (X) 44

45 5.7 (L ) (X, A)(A X ) X V 0,V 1 A σ : V 0 A V 1 A V =(V 0,V 1 ; σ) L (X, A) L (X, A) V, V φ i : V i V i φ 1 σ = σ φ 0 V = V L (X, A) L (X, A) E =(E 0,E 1 ; σ) E 0 = E 1 σ =id L (X, A) V, V E, E V E = V E L (X, A) [V 0,V 1 ; σ] L(X, A) L(X, A) 5.8 A = χ([v 0,V 1 ]) = [V 0 ] [V 1 ] χ : L(X, A) K(X, A) V =[V 0,V 1 ; σ] L(X, A) χ(v ) X 0 = X 0,X 1 = X 1 X [Ṽ ] [θn ]=[V 0 σ V 1 ] [V 1 id V 1 ] K(X 0 A X 1 ) X = X 0 K(X) Ṽ X 1 K(X/A) χ(v ) 5.9 X, Y p X : X Y X, p Y : X Y Y a b K(X) K(Y ) p X (a)p Y (b) K(X Y ) K(X) K(Y ) K(X Y ) K K K(X) K(Y ) K(X Y ) 5.10 X, Y X Y X Y 45

46 i) i : X Y X Y p : X Y X Y 0 K(X Y ) p K(X i Y ) K(X Y ) 0. ii) i K(X) K(Y ) K(X i Y ) K(X Y ) 5.11 (K ) K(X) K(Y ) K(X Y ) K (X, A) (Y,B) K K(X/A) K(Y/B) K((X/A) (Y/B)) = K((X Y )/(X B) (A Y )) K(X, A) K(Y,B) K(X Y,(X B) (A Y )) K 5.12 ( K ) X i K i (X) := K(S i X), (X, Y ) K i (X, Y ):= K i (X/Y ), X X + =(X, )( pt S i K i (X) K i (X, ) := K i (X + )=K(S i X, pt X) K i (pt) = K(S i ) 46

47 5.13 X, Y i, j K K(S i X) K(S j Y ) K((S i X) (S j Y )) K K i (X) K j (Y ) K i j (X Y ) K (pt) K (X) K (pt) (Bott K ) i) K (pt) ξ K 2 (pt) = K(S 2 ) K (pt) = Z[ξ]. ii) (X, A) Hausdorff ξ μ ξ : K i (X, A) K i 2 (X, A) i 5.15 (Bott K ) i) KO (pt) η KO 1 (pt), y KO 4 (pt), x KO 8 (pt), KO (pt) = Z[η, y, x]/ <2η, η 3,ηy,y 2 4x >. 47

48 ii) (X, A) Hausdorff x μ x : KO i (X, A) KO i 8 (X, A) i 5.16 ( K ) X X + X X + = X {pt} X K K cpt (X) := K(X + ), Kcpt i (X) :=K cpt(x R i ) (X, A)(A ) K K i cpt := K cpt((x A) R i ) 5.17 (K cpt Bott ) X K cpt (X) = K cpt (X C), KO cpt (X) = KO cpt (X R 8 ). ξ K cpt (C) = K(S 2 ) x KO cpt (R 8 ) = KO(S 8 ) 5.18 W = W 0 W 1 Z 2 C Cl n E k = D n W k n D n ( Cl n ) μ : E 0 S n 1 E 1 S n 1(S n 1 = D n ) μ(u, w) =(u, u w)( u =1) φ(w ):=[E 0,E 1 ; μ] K(D n,s n 1 ) 48

49 φ Z 2 C Cl n Grothendieck K φ : Mˆ n C K(D n,s n 1 ) i : R n R n+1 Grothendieck i : Mˆ n+1 C M ˆ n C i φ = 0 φ φ n : Mˆ n C /i Mˆ n+1 C K(Dn,S n 1 ) = K n (pt) Z 2 Clifford φ n : Mˆ n /i Mˆ n+1 KO(D n,s n 1 ) = KO n (pt) { Mˆ n C /i Mˆ n+1 C = Mn 1 C /i Mn C = Z n :even 0 n :odd Z n 0, 4(mod8) Mˆ n /i Mˆ n+1 = Mn 1 /i M n = Z 2 n 1, 2(mod8) (Atiyah-Bott-Shapiro ) φ : Mˆ C /i ˆ φ : Mˆ /i ˆ M C +1 M +1 = = K (pt), KO (pt) 49

50 5.1.2 Chern 5.20 (Splitting Principle) i) E X π : Y X a) π : H (X) H (Y ) b) π E π E = l 1 l n. X Y π ii) E X 2n π : Y X a) π : H (X) H (Y ) b) π E E k C = l k l k E k π E = E 1 E n. X Y π 5.21 ( ) Q f(0) = 1 f(x) Q[[x]] σ n x 1,,x n n f(x 1 ) f(x n )=1+F 1 (σ 1 )+F 2 (σ 1,σ 2 )+ F j (σ 1,,σ j )(x j j ) n n f(x) 50

51 5.22 A = {A k } a =1+ a 1 + a 2 + A {F k } F : A A F (a) =1+F 1 (a 1 )+F 2 (a 1,a 2 )+ F A F (ab) =F (a)f (b) ( Todd ) td(x) = x 1 e x =1+1 2 x x2 + Td Todd Chern c(e) Td C (E) =Td(c(E)) Todd n Td(X) :=Td n (TX)[X] X Todd 5.24 (  ) â(x) = x/2 sinh( x/2) = x x2 +   Pontrjagin c(e) Â(E) =Â(c(E))  a(x) =â(16x) A m =16 m  m A 5.25 E Td C (E C) =Â(E)2 51

52 5.26 ( L ) l(x) = x tanh( x) =1+1 3 x 1 45 x2 + ˆL HirzebruchL Pontrjagin c(e) ˆL(E) =ˆL(c(E)) L ˆl(x) =l(x/4) ˆL m =4 m L m ˆL 5.27 (Chern ) n E Chern c(e) Splitting Principle c(e) =1+c c n = n (1 + x k ) k=1 c j x j ch(e) =e x e xn = n + c 1 +(c 2 1 c 2)+ H 2 (X; Q) E Chern 5.28 Chern i) E,E X ch(e E )=ch(e)+ch(e ), ch(e E )=ch(e)ch(e ). ii) Hausdorff X Chern ch : K(X) H 2 (X; Q). 52

53 5.1.3 Clifford 5.29 X n E T (E) = r=0 j E I (E) v v+ <v,v>(v E x ) Cl(E) :=T (E)/I (E) E Clifford X Riemann T (X) Clifford X Clifford Cl(X) 5.30 E (p, q) P O (E) E O p,q - cl(ρ p,q ) O p,q R p,q ρ p,q Cl(R p,q ) cl(ρ p,q ):O p,q Cl(R p,q ) Cl(E) =P O (E) cl(ρp,q) Cl(R p,q ). E P SO (E) SO p,q - Cl(E) =P SO (E) cl(ρp,q) Cl(R p,q ) i) Clifford Cl(E) X Clifford a, b Cl(E x ) ab Cl(E x ) v, w E x Cl(E x ) vw + wv = 2 <v,w> 53

54 ii) Clifford Clifford α :Cl(E) Cl(E) α(v) = v(v E Cl(E)) α +1 1 Cl(E) Cl(E) =Cl 0 (E) Cl 1 (E) iii) e 1,,e n E x (< e i,e j >= η ij ) φ Cl(E x ) n L(φ) = e i φe j η ij i,j=1 Cl(E) L :Cl(E) Cl(E) iv) v 1 v k Λ k E x 1 sign(σ)v σ(1) v σ(k) k! σ S k E Clifford λ :Λ (E) = Cl(E) λ(λ even E)=Cl 0 (E), λ(λ odd E)=Cl 1 (E), λ(λ p E)={φ Cl(E) α L(φ) =(n 2p)φ} p =0,,n 54

55 5.32 X Riemann E Riemann Clifford Cl(E) Riemann Γ(Cl(E)) (φ ψ) =( φ) ψ + φ ( ψ). E e 1,,e n n e i = ω j i e j j=1 R(V,W) :Cl(E x ) Cl(E x ) R(V,W) R(V,W)(φ ψ) =(R(V,W)φ) ψ + φ (R(V,W)ψ) ( ) E X Riemann P SO (E) E SO n P SO (E) ξ : P Spin (E) P SO (E) P Spin (E) E P Spin (E) Spin n ξ(ug) =uξ 0 (g) u P Spin (E), g Spin n ξ 0 : Spin n SO n 55

56 5.34 ( ) X E E Stiefel-Whitney w 1 (E) =w 2 (E) =0 E H 1 (X; Z 2 ) 5.35 (Riemann ) Riemann X T (X) X Riemann X 5.36 ( ) Stiefel K ( ) E Riemann ξ : P Spin (E) P SO (E) M Cl(R n ) Spin n Cl 0 (R n ) Cl(R n ) Spin n μ : Spin n SO(M) M P Spin (E) S(E) :P Spin (E) μ M E (real spinor bundle) Cl(C n ) M C S C (E) :P Spin (E) μ M C E (complex spinor bundle) 5.38 Riemann E Clifford Cl(E) =P Spin (E) Ad Cl(R n ) S(E) Cl(E) 56

57 5.39 ( Riemann ) Riemann E Riemann S(E) Riemann i) E [e 1,,e n ] e i = n ω ji e j j=1 [e j ] S(E) [σ α ] σ α = 1 ω ji e i e j σ α 2 R(V,W) :S(E x ) S(E x ) R(V,W)σ = 1 <R(V,W)e i,e j >e i e j σ 2 i<j i<j ii) S(E) Cl(E) (φ σ) =( φ) σ + φ ( σ). R(V,W) Cl(E) Dirac 5.40 X Riemann S Cl(X) X Riemann S Riemann Dσ := n e j ej σ j=1 57

58 Γ(S) D :Γ(S) Γ(S) Dirac e 1,,e n T (X) D 5.41 X E m D : Γ(E) Γ(E) x X x k A α (x) :E x E x D = A α (x) α x α α m ξ = k ξ kdx k Tx (X) σ ξ (D) :E x E x σ ξ (D) :=i m A α (x)ξ α α =m D ξ 0 σ ξ (D) D 5.42 Riemann X Cl(X)- S Riemann <, > S X Dirac i) T (X) e S <eσ 1,eσ 2 >=< σ 1,σ 2 > e T x (X)s.t. <e,e>=1, σ 1,σ 2 S x. ii) φ Γ(Cl(X)) σ Γ(S) (φ σ) =( φ) σ + φ ( σ) X Riemann D X Dirac S Dirac D L 2 (S)(S Ker D =KerD 2 X Ker D 58

59 5.1.6 R n 5.44 ( ) m R p(x, ξ) R n R n α, α Dx α Dα ξ p(x, ξ) C α,α (1 + ξ )m α C α,α p(x, ξ) m Sym m 5.45 ( ) p Sym m u(x) ( R n Frechet ) u(x) =(2π) n/2 e i<x,ξ> û(ξ)dξ, Pu(x) =(2π) n/2 e i<x,ξ> p(x, ξ)û(ξ)dξ, P : R n m ΨDO m p Sym m P ΨDO m σ(p )=[p] Sym m /Sym m 1 P 5.46 P ΨDO m,q ΨDO l i) P s R P : L 2 s L2 s m ii) U R n iv) Q P ΨDO m+l. u U C (U) Pu U C (U). v) σ(p ) R n T (R n ) 59

60 5.47 K R n P ΨDO m u C0 supp(pu) K suppu K = Pu=0 P K ΨDO K,m 5.48 φ : U V R n U, V K U (φ P )u = P (u φ) φ 1 φ :ΨDO K,m ΨDO φk,m 5.49 ( ) τ : s, m R τ : L 2 s L2 s+m P, Q P Q 5.50 ( ) P ΨDO m p c>0 ξ c ξ p(x, ξ) p(x, ξ) 1 c(1 + ξ ) m P 5.51 P ΨDO m i) P Q ΨDO m PQ =Id S, QP =Id S. S, S ii) u L 2 s Rn U Pu U C (U) u U C (U). m >0 Pu = λu λ C u C (R n ) 60

61 ( ) i) X n E,F X Γ(E), Γ(F ) L 2 s(e),l 2 s(f ) Sobolev P : Γ(E) Γ(F ) s, m R P : L m s (E) L 2 s+m(f ) P ii) P :Γ(E) Γ(F ) P α ΨDO m P m ΨDO m (E,F) 5.53 E,F,G X P ΨDO m (E,F),Q ΨDO l (F, G) i) P s R P : L 2 s(e) L 2 s m(f ) ii) U X iv) Q P ΨDO m+l (E,G). u U C (U) Pu U C (U). v) φ : X X φ [(φ P )u] =P (φ u) φ :ΨDO m (φ E,φ F ) ΨDO m (E,F) 61

62 5.54 π : T (X) X X E,F T (X) π E,π F p T (X) Hom(π E,π F ) p Sym m p m Sym m (E,F) 5.55 P ΨDO m (E,F) σ(p ) Sym m (E,F)/Sym m 1 (E,F) 5.56 ( ) P ΨDO m (E,F) σ(p ) T (X) X Riemann C p(ξ) 1 C(1 + ξ ) m P 5.57 (Fredholm ) T : H 1 H 2 ran T Ker T Coker T Fredholm T )=dim(kert ) dim(coker T ) 5.58 P ΨDO m (E,F) X i) Q ΨDO m (E,F) PQ =Id S, QP =Id S S, S ii) u L 2 s (E) X U Pu U C (U) u U C (U). 62

63 iii) s R P Fredholm P : L 2 s (E) L2 s m (E) s 5.59 P :Γ(E) Γ(E) Riemann X m i) Γ(E) L 2 (E) Γ(E) =KerP Im P ii) H :Γ(E) Ker P Green Q ΨDO m (E) PG = GP =Id H. iii) m>0 P λ E λ d(λ) := dim λ Λ E λ d(λ) cλ n(n+2m+2)/2m c {E λ } L 2 (E) Atiyah-Singer F = F (H 1,H 2 ) Fredholm F Z F π 0 (F ) Z 63

64 5.61 P P ) σ(p ) 5.62 ( ) X E F P :Γ(E) Γ(F ) TX DX σ(p ) DX = SX E F σ(p ) K cpt (TX) σ(p ):=[π E,π F ; σ(p )] K cpt (TX) = K(DX, SX). X f : X R N X R N N f : X N Thom K cpt (TX) K cpt (TN) i! : K cpt (TN) K cpt (T R N ) f! : K cpt (TX) K cpt (T R N ) T R N pt Thom q! : K cpt (KR n ) K(pt) = ZR P top index(p ) top index(p ):=q! f! σ(p ) Z ( Atiyah-Singer ) n X P i) ii) P )=top index(p ). P =( 1) n {ch(σ(p )) π Â(X) 2 }[TX]. 64

65 iii) X P =( 1) n(n+1)/2 {π! ch(σ(p )) Â(X)2 }[X] (Euler ) Clifford X Riemann S S =Cl(X) =Cl 0 (X) Cl 1 (X). Cl(X) Dirac D 0 :Γ(Cl 0 (X)) Γ(Cl 1 (X)) d + d :Γ(Λ even (X)) Γ(Λ odd (X)) H D 0 =dimh even dim H odd = χ(x) 5.65 ( ) X 4k Riemann Clifford ω C =( 1) k ω S =Cl(X) =Cl + (X) Cl (X) Dirac Cl(X) Dirac D + :Γ(Cl + (X)) Γ(Cl (X)) X H 2k (X; R) D + =dim(h 2k ) + dim(h 2k ) =sig(x). 65

66 Atiyah-Singer L L(X) =sig(x). E 2m X D + E :Γ(Cl+ (X) E) Γ(Cl (X) E) (D + E )={ch 2(E) L(X)}[X] ch 2 (E) := k 2 k ch k E (Atiyah-Singer  ) X 2m Riemann /S C Dirac D /S C /S C = /S + C /S C /D + :Γ(/S + C ) Γ( /S C ) X  ( /D + )=Â(X). E X /D + E :Γ(/S+ C (X) E) Γ( /S C (X) E) ( /D + E )={ch(e) Â(X)}[X]. 66

67 [1] D. Husemoller: Fibre Bundles, 3rd edition (Springer, 1993). [2] H.B. Lawson, Jr. and M-L. Michelsohn: Spin Geometry (Princeton Univ. Press, 1989). 6 [LastUpdate: ] ( Grassmann ) 1) R n+k n (n ) V n,k = V n (R n+k ) V n,k R n(n+k) V n,k R n(n+k) Stiefel V n,k C n(n+k) V n,k V k,n C 2) R n+k n G n,k = G n (R n+k ) π : V n,k G n,k G n,k nk Grassmann G n,k G k,n C 3) E(γ n k )={ (P, v) P G n,k,v P R n+k} π : E(γ n k ) (P, v) P G n,k γ n k =(E(γn k ),G n,k,π) G n,k n G n,k n 67

68 4) R n+k R n+k+1 G n,k G n,k+1 (G n,k ) k 0 G n = G n, = G n (R ) Grassmann γ n k γn k+1 γn E(γ n )={(P, v) P G n,v P R } G n R 6.2 ( Grassmann ) 1) R n+k n G n,k = G n (R n+k ) π : V n,k G n,k G n,k nk Grassmann G n,k G k,n C G n,k G n,k 2) E( γ k {(P, n )= v) P G } n,k,v P R n+k π : E( γ n k ) (P, v) P G n,k γ n k =(E( γn k ), G n,k,π) G n,k n G n,k n 3) ( G n,k ) k 0 G n = G n, = G n (R ) Grassmann γ k n γn k+1 γn { E( γ n )= (P, v) P G } n,v P R G n R 68

69 6.3 ( ) 1) ξ n =(E,B,π) Hausdorff B n f : B G n ξ n = f γ n B n B G n G n Hausdorff n γ n 2) G n Hausdorff n γ n ( Grassmann ) 1) C n+k n (n ) Vn,k C = V n (C n+k ) Vn,k C Cn(n+k) Vn,k C Cn(n+k) Stiefel V n,k C C 2n(n + k) Vn,k C V k,n C C 2) C n+k n G C n,k = G n(c n+k ) π : V C n,k GC n,k GC n,k 2nk Grassmann G C n,k GC k,n C 69

70 3) E(γ n,c k )= { (P, v) P G C n,k,v P C n+k} π : E(γ n,c k ) (P, v) P G C n,k γ n,c k =(E(γ n,c k ),G C n,k,π) GC n,k n G C n,k n 4) C n+k C n+k+1 G C n,k GC n,k+1 (G C n,k ) k 0 G C n = G C n, = G n (C ) Grassmann γ n,c k γ n,c k+1 γn,c E(γ n,c )= { (P, v) P G C n,v P C } G C n C 6.5 ( ) ω n =(E,B,π) Hausdorff B n f : B G C n ωn = f γ n,c B n B G C n G C n Hausdorff n γ n,c ( ) 70

71 1) c 1,,c n Grassmann G n (C ) Chern H (BU(n)) = H (BGL(n, C)) = Z[c 1,,c n ], (12) H (BSU(n)) = H (BSL(n, C)) = Z[c 2,,c n ]. (13) 2) q 1,,q n Pontryagin H (BSp(n)) = Z[q 1,,q n ]. (14) 3) w 1,,w n Stiefel-Whitney p 1,,p n Pontryagin e Euler K 2 2 K 2 H (BO(n); K 2 )=H (BGL(n, R); Z 2 )=K 2 [w 1,,w n ],(15) H (BSO(n); K 2 )=H (BSL(n, R); Z 2 )=K 2 [w 2,,w n (16) ], H (BSO(2m +1);K) =K[p 1,,p m ], (17) H (BSO(2m); K) =K[p 1,,p m 1,e]. (18) Poincaré-Hopf 6.7 ( ) X n C M C 1) X U p U e a =(e 1,,e n ) e a M T (M) U (π, π ):T (U) U T p (M) p D n V ( U) X V T (M) π g :(V,V p) (T p (M),T p (M) p) p M T p (M) g g : H n (V,V p) H n (T p (M),T p (M) p) 71

72 e a V H n (V,V p) = H n 1 ( V ) = Z u n U H n (T p (M),T p (M) p) = Z u n T p(m) X p Ind(X, p) g u n = Ind(X, p) u n 2) X M p 1,,p k X Ind(X) k Ind(X) = Ind(X, p j ) j=1 6.8 (Poincaé-Hopf) M n C X M C M i) X M M ii) X M Ind(X) =χ(m). Proof. [ ( 1992)] 1) M R n n U U W Gauss g : U p W p / W p S n 1 g : H n 1 ( U) H n 1 (S n 1 ) g [ U] =Ind(X)[S n 1 ] Ind(X) U 72

73 2) M M R m M R m N(M) Y i) Y M X N(M) M ii) Y N(M) iii) X p j Ind(X, p j ) = Ind(Y,p j ) 3) (M;, M) Morse f Riemann f Ind( f) =χ(m) 4) 1) 2) Ind(X) = Ind(Y ) N(M) X 3) Ind(X) =χ(x) Euler Thom 6.9 ξ =(E,B,π) n B ξ E B ξ ξ 0 =(E 0,B,π) 6.10 (Thom Thom ) ξ =(E,B,π) n j b : F = R n F b = π 1 (b) E F F b U H n (F, F 0 ; Z) F H n (F, F 0 ; Z) (E,E 0 ) H (E,E 0 ; Z) : H (E; Z) H (E,E 0 ; Z) H (E,E 0 ; Z) 73

74 i) H i (E,E 0 ; Z) =0(i<n). ii) H n (E,E 0 ; Z) U(ξ) b B j b (U(ξ)) = U U(ξ) ξ Thom iii) φ : H i (B; Z) α φ(α) =π (α) U(ξ) H i+n (E,E 0 ; Z) Thom iv) Thom n ξ =(E,B,π),ξ = (E,B,π ) f : ξ ξ f : B B U(ξ) =U(f 1 ξ )= f (U(ξ )) 6.11 (Euler ) ξ =(E,B,π) n ξ Thom U(ξ) j :(E, ) (E,E 0 ) e(ξ) =(π ) 1 j (U(ξ)) H n (B; Z) ξ Euler 6.12 (Gysin ) ξ =(E,B,π) n (E,E 0 ) H q 1 (E 0 ) δ H q (E,E 0 ) j H q (E) i H q (E 0 ) Thom (Thom-)Gysin H q 1 (E 0 ; Z) φ 1 δ H q n (B; Z) e(ξ) H q (B; Z) (π E 0 ) H q (E 0 ; Z). 74

75 6.13 (Euler ) ξ =(E,B,π),ξ =(E,B,π ) n i) Euler f : B B f 1 ξ B ii) e(ξ) =φ 1 (U(ξ) U(ξ)). e(f 1 ξ)=f (e(ξ)). iii) ξ ξ e(ξ )= e(ξ). iv) n 2e(ξ) =0. v) ξ e(ξ) =0. vi) ξ ξ ξ ξ =(E E,B B,π π ) j b j b : F F π 1 (b) π 1 (b ) e(ξ ξ )=e(ξ) e(ξ ), e(ξ ξ )=e(ξ) e(ξ ) ( Euler ) M n n C M n Euler M n Euler e(m n )=e(τ(m n )). M n χ(m n ) M n Euler [M n ] M n χ(m n )= e(m n ), [M n ] 75

76 6.15 ( Euler ) n C M n n + k Euclid R n+k M n ν k e(ν k )= (CP n ) i) γ 1,C =(E(γ 1,C ), CP,π) Euler α H 2 (CP ; Z) ii) γ 1,C k H (CP ; Z) =Z[α]. γ 1,C ι : CP k CP H (CP k ; Z) =Z[ι (α)]/(ι (α) k+1 =0) Z 2 -Euler Thom 6.17 (Thom Thom ) ξ =(E,B,π) n j b : F = R n F b = π 1 (b) E U H n (F, F 0 ; Z 2 ) (E,E 0 ) Z 2 H (E,E 0 ; Z 2 ) : H (E; Z 2 ) H (E,E 0 ; Z 2 ) H (E,E 0 ; Z 2 ) i) H i (E,E 0 ; Z 2 )=0(i<n). ii) H n (E,E 0 ; Z 2 ) U (ξ) b B j b (U (ξ)) = U U (ξ) ξ Z 2 -Thom Z 2 76

77 iii) φ : H i (B; Z 2 ) α φ(α) =π (α) U (ξ) H i+n (E,E 0 ; Z 2 ) Z 2 -Thom iv) Z 2 -Thom n ξ =(E,B,π),ξ =(E,B,π ) f : ξ ξ f : B B U (ξ) =U (f 1 ξ )= f (U (ξ )) 6.18 (Z 2 -Euler ) ξ =(E,B,π) n ξ Z 2 -Thom U (ξ) j :(E, ) (E,E 0 ) e (ξ) =(π ) 1 j (U (ξ)) H n (B; Z 2 ) ξ Z 2 -Euler 6.19 (Gysin ) ξ =(E,B,π) n (E,E 0 ) H q 1 (E 0 ) δ H q (E,E 0 ) j H q (E) i H q (E 0 ) Thom (Thom-)Gysin H q 1 (E 0 ; Z 2 ) φ 1 δ H q n (B; Z 2 ) e (ξ) H q (B; Z 2 ) (π E 0 ) H q (E 0 ; Z 2 ) (Z 2 -Euler ) ξ =(E,B,π),ξ =(E,B,π ) n 77

78 i) Z 2 -Euler f : B B f 1 ξ B e (f 1 ξ)=f (e (ξ)). ii) e (ξ) =φ 1 (U (ξ) U (ξ)). iii) ξ e (ξ) =0. iv) e (ξ ξ )=e (ξ) e (ξ ), e (ξ ξ )=e (ξ) e (ξ ) ( Z 2 -Euler ) M n n C M n Z 2 -Euler M n Z 2 -Euler e (M n )=e (τ(m n )). M n χ(m n ) M n Euler [M n ] M n Z 2 χ(m n ) e (M n ), [M n ] mod (RP n Z 2 ) i) γ 1 =(E(γ 1 ), RP,π) Z 2 -Euler ˆα H 1 (RP ; Z 2 ) H (RP ; Z 2 )=Z 2 [ˆα]. ii) γ 1 k γ1 ι : RP k RP H (RP k ; Z 2 )=Z 2 [ι (ˆα)]/(ι (ˆα) k+1 =0). 78

79 6.23 ( Z 2 -Euler ) n C M n n + k Euclid R n+k M n ν k e (ν k )= Stiefel-Whitney 6.24 (Stiefel-Whitney ) Hausdorff Stiefel-Whitney (SW I) ξ =(E(ξ),B(ξ),π) w i (ξ) H i (B(ξ); Z 2 )(i =0, 1, 2, ) w 0 (ξ) =1 H 0 (B(ξ); Z 2 ) ξ n w i (ξ) =0(i>n) w i (ξ) ξ i Stiefel- Whitney w(ξ) =1+w 1 (ξ)+w 2 (ξ)+ H (B(ξ); Z 2 ) Stiefel-Whitney (SW II) ( f : ξ η f : B(ξ) B(η) f w(ξ) =f (w(η)). (SW III) (Whitney ξ ξ w(ξ ξ )=w(ξ) w(ξ ). (SW IV) G 1,1 = RP 1 γ 1 1 ˆα H1 (RP 1 ; Z 2 ) = Z 2 w 1 (γ 1 1 )=ˆα H1 (RP 1 ; Z 2 ). 79

80 (SW V) ξ n w n (ξ) ξ Z 2 -Euler w n (ξ) =e (ξ) (Stiefel-Whiney ) (SW I)-(SW V) Stiefel-Whitney 6.26 ( ) n G n Z 2 H (G n ; Z 2 ) γ n Stiefel-Whitney Z 2 H (G n ; Z 2 ) = Z 2 [w 1 (γ n ),,w n (γ n )] 6.27 ( ) ξ Hausdorff ξ w 1 (ξ) = ( ) ξ Hausdorff n ξ q w n (ξ) =w n 1 (ξ) = = w n q+1 (ξ) = ( Stiefel-Whitney ) RP k γk 1 Z 2-Euler ˆα H 1 (RP k ; Z 2 ) w(rp k )=(1+ˆα) k+1. 80

81 6.2.5 Chern 6.30 (Chern ) Hausdorff Chern (C I) ω =(E(ω),B(ω),π) c i (ω) H 2i (B(ω); Z) (i =0, 1, 2, ) c 0 (ω) =1 H 0 (B(ω); Z) ω n c i (ω) =0(i>n) c i (ω) ω i Chern c(ω) =1+c 1 (ω)+c 2 (ω)+ H (B(ω); Z) Chern (C II)( f : ω θ f : B(ω) B(θ) f c(ω) =f (c(θ)). (C III) (Whitney ω ω c(ω ω )=c(ω) c(ω ). (C IV) G C 1,1 = CP 1 γ 1,C 1 α H 2 (CP 1 ; Z) = Z c 1 (γ 1,C 1 )=α H 2 (CP 1 ; Z). (C V) ω n c n (ω) ω ω R Euler c n (ω) =e(ω R ). 81

82 6.31 (Chern ) (C I)-(C V) Chern 6.32 ( ) n G C n H (G C n ; Z) γn,c Chern Z H (G C n ; Z) = Z[c 1 (γ n,c ),,c n (γ n,c )] 6.33 ( ) ω Hausdorff n ω q c n (ω) =c n 1 (ω) = = c n q+1 (ω) = ( Chern ) CP k γ 1,C k Euler α H 2 (CP k ; Z) c(cp k )=(1+α) k Pontrjagin 6.35 (Pontrjagin ) ξ = (E,B,π) Hausdorff B n ξ ξ C Chern c 2j (ξ C) ξ Pontrjagin p j (ξ) p j (ξ) =( 1) j c 2j (ξ C) H 4j (B; Z) (j =0, 1, 2, ). H (B; Z) p(ξ) p(ξ) =1+p 1 (ξ)+ ξ Pontrjagin 82

83 6.36 (Pontrjagin ) Pontrjagin (P I) ξ =(E(ξ),B(ξ),π) ξ Pontjagin P i (ξ) H 4i (B(ξ); Z) (i =0, 1, 2, ) p 0 (ξ) =1 H 0 (B(ξ); Z) ξ n i >[n/2] p i (ξ) =0 (P II)( f : ξ η f : B(ξ) B(η) f p(ξ) =f (p(η)). (P III) (Whitney ξ ξ p(ξ ξ )=p(ξ) p(ξ ) moda. A H (B(ξ); Z) 2 (P IV) ξ 2n p n (ξ) =e(ξ) ( ) Λ 1/2 i) 2n +1 G 2n+1 Λ H ( G 2n+1 ;Λ) γ 2n+1 Pontrjagin Λ H ( G 2n+1 ;Λ) = Λ[p 1 ( γ 2n+1 ),,p n ( γ 2n+1 )] 83

84 i) 2n G 2n Λ H ( G 2n ;Λ) γ 2n Pontrjagin Euler Λ H ( G 2n ;Λ) = Λ[p 1 ( γ 2n ),,p n 1 ( γ 2n ),e( γ 2n )] p n ( γ 2n )=e( γ 2n ) (Chern ) n ω Chern ω R Pontrjagin 1 p 1 (ω R )+p 2 (ω R ) +( 1) n p n (ω R )=c(ω)c( ω). ω ω c( ω) =1 c 1 (ω)+c 2 (ω) +( 1) n c n (ω) (Euler ) ξ CW B n ˆξ n B n 1 ˆξ n o(ˆξ) H n (B; Z) ξ Euler e(ξ) 6.40 (Stiefel-Whitney ) ξ CW B n B q 1 n q +1 q Z 2 - o q (ξ) H q (B; Z 2 ) ξ Stiefel-Whitney w q (ξ) 84

85 6.41 (Chern ) ω CW B n B 2q 1 n q +1 2q o q (ω) H 2q (B; Z) ω Chern c q (ω) 6.42 ( ) 1) ξ CW B ξ w 1 (ξ) =0 ( B Haussdorf ) ξ H 0 (X; Z 2 ) 2) ω CW n ω SU(n) c 1 (ω) =0 Proof. 1) P (G, B) P (H, B) P (G, B) (P/H,B,G/H) ξ ξ P (O(n),B) P (SO(n),B) P O(n) SO(n) O(n)/SO(n) = Z 2 Z 2 B 1 B B (1) ξ B B (1) ξ B (1) SO(n) SO(n) B (1) B (1) ξ n B (1) ξ n B (1) ξ o(ξ) H 1 (B;O(n)/SO(n)) = H 1 (B; Z 2 ) w 1 (ξ) B Z 2 85

86 B Z 2 H 0 (B; Z 2 ) 2) B (2) SU(n) π j (U(n)/SU(n)) = π j (S 1 )= 0(j 2) B B (2) B (2) SU(n) SU(n) π 1 (SU(n)) = 0 ω B (2) SU(n) B (2) ω P (U(n),B) o 1 (ξ) =c 1 (ξ) H 2 (B; Z) ( ) ξ CW X n ξ 3 i) ɛ X 1 X (j) X j ξ k ξ ɛ k X (1) σ X (2) ξ σ n 3 k =0 n =2 k =1 n =1 k =2 ii) ξ O(n) SO(n) P P Spin(n) P ξ p : P P Spin(n) Ẽ(P ) X λ SO(n) p E(P ) = X (19) 86

87 iii) ξ O(n) SO(n) P σ H 1 (E(P ); Z 2 ) H 1 (SO(n); Z 2 ) ξ σ ( ) ξ CW X n 1) ξ w 1 (ξ) =w 2 (ξ) =0 2) ξ σ H 1 (E(P ); Z 2 ) i (σ) =1 σ H 1 (X; Z 2 ) 0 H 1 (X; Z 2 ) π H 1 (E(P ); Z 2 ) i H 1 (SO(n); Z 2 ) = Z 2 0 (20) 7 Knots and Links [LastUpdate: ] ( ) c[l] def = 87

88 (linking number) 7.2 ( ) D c sign(c) =+1, sign(c) = ( ) K 1,K 2 Link(K 1,K 2 ) def = 1 sign(c). 2 c K 1 K (bridge index) 7.4 ( ) br def =height local maximum points br[l 1 L 2 ]=br[l 1 ]+br[l 2 ] Schubert S(α, β) S(α, β) : gcd(α, β) =1, α <β<α, β: 7.5 (i) 2- S(α, β) S(α,β ) α = α, β ±1 β (modα). (ii) 2- (i) α = α, β ±1 β (mod2α). 88

89 (braid index) 7.6 ( ) b[l] def =braid b[k 1 K 2 ]=b[k 1 ]+b[k 2 ] (unknoting number) 7.7 ( ) u[l] def = 7.2 Seifert 7.8 (Seifert ) S 3 L F F = L L Seifert 7.9 (Seifert ) L S 3 F L Seifert f : F [ 1, 1] S 3 S 3 F f + (x) =f(x, 1),f = f(x, 1) F c 1,c 2 c + 1 = f + (c 1 ) c 2 = f (c 2 ) S 3 L(c + 1,c 2 ) c 1,c 2 φ : H 1 (F ) H 1 (F ) Z L Seifert F Seifert H 1 (F ) φ Seifert 89

90 7.10 (S- ) V,W W = 1 x u 0 v V W V V W W T V T V T W T S L Seifert S (genus) 7.12 ( ) g[l] def = L Seifert (signature) L Seifert F, F Seifert M F 7.13 ( ) b : H H Z H = Z ( Z ) b 0 1 = ( ) (H, b), (H,b ) 90

91 7.15 L Seifert F M F + M T F 7.16 ( ) σ[l] def =sign(m F + M F T ). σ[l 1 L 2 ]=σ[l 1 ]+σ[l 2 ], σ[±l ]= σ[l]. K σ[k] 2u[K]. (21) 7.17 ( ) n[l] def =(dim rank)(m F + M F T ). n[l] r 1 (r = L ) ( ) G[L] :=π 1 (E); E := S 3 N(L). 91

92 7.19 ( ) γ : G = π 1 (E) H 1 (E) = Z r Hurwitz (r = L ) kernel Ker γ =[G, G] E p : E γ E H 1 (E) t 1,,t r ZH 1 (E) Laurent Λ=Z[t 1,,t r ] H 1 (E) = π 1 (E)/π 1 (E γ ) E γ H 1 (E γ ) Λ- (1) L =Λ- H 1 (E γ ). (2) L Alexander A[L] =Λ- H 1 (E γ,p 1 (e))(e E). 7.4 L D c L + (D, c) L (D, c) L 0 (D, c) Alexander-Conway Skein 7.21 (Alexander-Conway ) L L (z) Z[z] (AC0) L L L = L (AC1) =1. (AC2) L+ L = z L0. 92

93 : L = L 1 L 2,L 1 L 2 = L = (1 Alexander ) L Seifert F F Seifert M F Δ L (t) def = ±t m det(m F tm F T )=a 0 + a k t k (a 0 > 0) (22) F 1 Alexander Alexander-Conway Δ L (t). = L (t 1/2 t 1/2 ). M n L S 3 n 1 Alexander Δ L (t) 1 n ω n H 1 (M n ) = Δ L (ω k ). = 0 =0 k= (1) M Λ- m, n Λ m Λ n M 0 Λ m Λ n Λ (m, n) P M (2) d P (n d) Λ E d (M) M d Λ E d (M) Δ d (M) M d 93

94 7.24 ( Alexander ) L H 1 (E γ ) Alexander A(L) d Alexander : Δ (d) L Alexander : Δ L def =Δ (0) L def =Δ d+1 (A(L)) = Δ d (H 1 (E γ )), : r L = K 1 K r i) Δ L (t 1,,t r ) =Δ. L (t 1 1,,t 1 r ). ii) L = L K r, λ i =Link(K i,k r ) r =2 Δ L (t 1,,t r 1, 1) =. tλ t 1 1 Δ L (t 1), r>2 Δ L (t 1,,t r 1, 1). =(t λ 1 1 t λ r 1 r 1 1)Δ L (t 1,,t r 1 ).. = (mod ) 1 Alexander r >1 Δ L (t) =(t 1)Δ L (t,,t) Jones Skein 7.25 (Jones ) L 1 V L (t) Z[t 1/2,t 1/2 ] (J0) L L V L = V L (J1) V (t) =1. (J2) t 1 V L+ (t) tv L (t) =(t 1/2 t 1/2 )V L0 (t). 94

95 (L) L M n L S 3 n i) V L (1) = ( 2) (L) 1, ii) V L ( 1) = L (2i), iii) V L (e 2πi/3 )=1, { iv) V L (i) = 2 ( L 1)/2 ( 1) Arf(L) (Arf(L) ) 0 ( ) v) V L (e πi/3 )=±i (L) 1 ( 3i) rankh 1(M 2 (L);Z 3 )., L 1 K L λ =Link(K, L K) V L (t) =t 3λ V L (t) State L c c A B c A R + (c) B R (c) 7.26 (Kauffman ) A, B, d K = K (A, B, d) = σ L K σ d σ Kauffman σ L {+, } σ c R σ(c)(c)l 1 K σ (AR + (c)+br (c)) = K σ R σ(c) (c) c σ c A, B 95

96 7.27 L w(l) L L (A) :=( A 3 ) w(l) K (A, A 1, A 2 A 2 ) Laurent Jones V L (t) =L L (t 1/4 ) Homfly Skein 7.28 (Homfly ) L 2 P L (α, z) Z[α, α 1,z,z 1 ] (H0) L L P L = P L (H1) P =1. (H2) αp L+ α 1 P L = zp L0. L (z) =P L (1,z), V L (t) =P L (t, t 1/2 t 1/2 ). 96

97 L M n (L) Jones i) P L (a, z) =P L (a, z), ii) P L (a, z) =P L ( a 1,z), iii) P L1 L 2 (a, z) =P L1 (a, z)p L2 (a, z), iv) P L1 +L 2 (a, z) = a 1 a P L1 (a, z)p L2 (a, z), z v) P L (a, a 1 a) =1, vi) P L ( a, z) =P L (a, z), vii) P L (a, z) =P L ( a, z) =( 1) (L) 1 P L (a, z), viii) P L (i, i) =( 2i) rankh 1(M 3 (L);Z 2 ) Q Skein L L ± L ± 2 L 0 L (Q- ) L 1 Q L (x) Z[x, x 1 ] (Q0) L L Q L = Q L. (QI) Q (x) =1. (QII) Q L + (x)+q L (x) =x{q L 0 (x)+q L 1 (x)}. i) Q L (1) = 1, ii) Q L ( 1) = ( 3) rankh 1(M 2 (L);Z 3 ), iii) Q L (2) = L (2i) 2, iv) Q L ( 2) = ( 2) (L) 1. 97

98 7.4.5 Kauffman Skein 7.30 (Kauffman ) L 2 Λ L (a, x) Z[a, a 1,z,z 1 ] Λ L (K0) Λ Λ Λ L =Λ L. (K1) Λ (α, z) =1. (K2) Λ L + (α, z)+λ L (α, z) =z{λ L 0 (α, z)+λ L 1 (α, z)}. (K3) Λ T+ = αλ D, Λ T = α 1 Λ D. T ± L D L w(l) L F L (α, z) :=α w(l) Λ L (α, z) 2 F L Kauffman Q L (z) =F L (1,z), V L (t) =F L ( t 3/4,t 1/4 + t 1/4 ). 7.5 Yang-Baxter ( ) T i j k l δj i i j i j T k l 98

99 7.32 ( ) 7.33 L (+) ( ) L (+) a b c d = Rab ( ) a b c d = cd, ab R cd L T (L) T (L) R i) (channel unitarity) R ab ij Rij cd = δa c δb d. ii) (cross-channel unitarity) Rjb ia jd R ic = δa c δd b. iii) (Yang-Baxter ) R ab ij Rjc kf Rik de = Rbc ij Rai dk Rkj R ab ij R jc kf R ik de = R bc ij R ai dk ef, R kj ef. 99

100 7.35 R R ab cd = Aδa c δb d + A 1 δ ab δ cd, R ab cd = A 1 δ a c δb d + Aδab δ cd n A n = A 2 A 2 T (L) A Kauffman K 7.36 L M ab, M ab. R ab cd, R ab cd. δb a 7.37 L τ( L ) M ab,m ab,r, R τ( L ) i) ( M ai M ib = δ a b. ab ii) R cd = M cirdj iam jb. 100

101 ab iii) II R ij Rij cd = δa c δb d. iv) R, R Yang-Baxter 7.38 M =(M ab ),R ( ) 0 ia M = Aσ 2 =, ia 0 R = AM 1 M + A 1 I d =TrM(M T ) 1 = A 2 A 2 τ(l) Kauffman τ(l) =d K. Kauffman Yang-Baxter A = 1(d = 2) τ(l) Penrose 101

102 [Sma60] Smale, S.: Bull. Amer. Math. Soc. 66, (1960). [Sma61] Smale, S.: Generalized Poincaré conjecture in dimensions greater than four, Ann. Math. 74, (1961). [Sta60] Stallings, J.: Polyhedral homotopy spheres, Bull. Amer. Math. Soc. 66, (1960). 102