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 F T n := {( x, l) F n+1 P n F x l}. T n P n F. tautological 3. M T M. f : M N C 2 -. T f : T M T N 4. C 1 - M T M ( M R N ), T M M. 5. p : E N C 1 -. x E T p x : T x E T f(x) N ( submerssion ) b N, f 1 (b) R dime dimn., p : E N, N C 1 - p : E N, p 6. M M. 7. p : E X, p E B. 8. n- M, n E M. M C 1 -, E T M ( ). 2

(4/19) 2, ( ) 9. S 2 10. S 2m+1 S n S n S n ; id S n x x 11. S 5 12. Lie G T G G R dimg ( ) 13. S 7 S 7 R 7 (S 7 Lie ) 14. E, F X, f : E F. x X, f x : E x F x. Ker(f) = {u E u E x, f x (x) = 0, x X} E, Ker(f) E, F X, f : E F., f Ex : E x F x x U. 15. (Faldbau ) I [0,1] n- I n I n, 2 16. 17. (Swan [ ] )Swan X C 0 (X) R C 0 (X)- P E X C 0 (X)- P = Γ(E) Γ(E) 3

(4/26) 18. E X, F X, F E E/F, E, F = E/F 19. E B, s 1,..., s n : B E b B, s 1,..., s n s 1,..., s n E b n, n- I E = I E/I 20. E X n n- S n E, n- n E,, E n = S n E n E 21. E n T n E n T n RP n R n+1 22. ξ = η 1 η n n Whitney r n r (η 1 η n ) = i(1)< <i(r) λ i(1) λ i(r). ( : B 5.6 23. E X, F X, G X, i : E F p : F G b X. i 0 E x i x x Fx Gx 0., F = E G 24. X X E, E E () X 25. S 1 ϵ, η S 1 n ϵ n ϵ n 1 η ϵ ϵ = η η 2.1 ( ) 26. X Wikipedia 27. 28. Stone 29. 4

(5/10) 5 24 30. E X m. m- m E, E n P n R 31. f : A B g : B C E, E B (g f) (E) = g (f (E)), f (E E ) = f (E) f (E ), f (E E ) = f (E) f (E ) 32. M S 1 Mobious f : S 1 S 1 ; z z n n f (M) n f (M) Mobious 33. C - g : M N T M (f) := {u x X, u T x (g 1 (g(x)))} T M T M = T M (f) g (T N). 34. n p : E B, P (E) f : P (E) B b B, f 1 (b) m E P (E) m 1 η 1,..., η m f (E ) = η 1 η m 35. C - g : M N U g(m) T M T N U 2.2 ( 3 4 ) 36 (Faldbau ). B 1 = A [0, c] B 2 = A [c, 1] 0 < c < 1 B = A [0, 1] p : E B E E B1 E B2 E 37. E A [0, 1] A {U i i I} E E Ui [0,1] 38. A 1 A 2 B 1 B 2 A, B A B A 1 B 1 A 2 B 2 39. A 1 A 2 A 5

(5/17) 40. R G n,k l G n,k π l : R n l U l := {l G n,k π l (l ) n } {U l l G n,k } G n,k C ω - 41. R G n,k {A Mat(n n; R) A 2 = A, t A = A, ranka = k } Mat(n n; R) G n,k Mat(n n; R) 42 (, Hatcher ). χ B B = i I U i χ Ui {W j j 1} χ Wj n N b B #{ i I b U i } n #{W j } n 43. p : E(η) B η g : E R Gauss β : E(η η) R β(x, y) = g(x) g(y) η η 44. R G n,k = Gn,n k 45. R 46. E n (R k ) G n,k 47 ( Grassman ). G n,k k l R n V n (R k ) G n,k G n,k 2:1 48 (Plüker ). V k (R n ) n R k (x 1,..., x n ) x 1 x n G n,k RP k C n 1 49. F R C H η FP η η FP FP 1 FP FP FP FP F R C 50. q F q k > n (F q ) k n (qk 1)(q k q) (q k q n 1 ) (q n 1)(q n q) (q n q n 1 ) 51. γ FP n T FP n, ϵ T FP n ϵ = γ γ 6

(5 31,.,...,,.. 3.,.,. 7

F1 Q (6/14)., ( ). ( ) Allen Hatcher, Vector bundle and K-theory ( HP ) 8

(6/14) 52. C i (X; R) C j (X; R) C i+j (X; R) H (X; R) = n H n (X; R) 53. T 3 = S 1 S 1 S 1 3 S 1 CW 0-1- 1. T 3 CW 2. H k (T 3 ; Z). 3. H (T 3 ; Z) 54. X, Y, X Y X = S 2 S 4, Y = CP 3 X = S 2 S 2, Y = CP 2 CP 2 55. Σ g g H 1 (Σ; Z) H 1 (Σ; Z) Z 56. S 1 S 1 A = S 1 {x} B = {y} S 1 A B A B 57. M n N M n 1 M \ N N N S n n 1 N 58. H n 1 (X; R) R H n (X; R) Hom R (H n (X; R), R) (X, A) 59. M n K M H i (M, M K; Z) i > n α H n (M, M K; Z) x K, ρ x (α) H n (M, M \ {x}) ρ x (M, M \ K) (M, M \ {x}) 60 ( ). K S n x K, y S n \ K : H i 1 (K, {x}; Z) = H n i (S n \ K, {y}; Z). 9

(6/21) 61. H (RP 2k ; Z) = Z[α]/(2α, α k+1 ), degα = 2 H (RP 2k+1 ; Z) = Z[α, β]/(2α, α k+1, β 2, αβ), degα = 2, β = 2k + 1 62. (1) S k+l S k S l H k+l (S k+l ; Z) H k+l (S k S l ; Z) (2) C n+1 C n+1 ; (z 1,..., z n+1 ) (z d 1,..., z d n+1) f d : C n P C n P f d : H2n (C n P ; Z) H 2n (C n P ; Z) 63. X CP /CP n 1 H (X; Z) Cohen-Macauley 64. M 2 H k (X; Z) H n k (X; Z) H n (X; Z) = Z. 65. X 3 H (X; Z/m) 66 (Künneth ). X, Y CW H i (X; R) R Y i+j=n H i (X; R) H j (Y ; R) H n (X Y ; R) 67 (Hopf ). f : S 4n 1 S 2n C(f) C(f) := S 2n f e 4n 2n- 4n- f H 2n (C(f); Z) = Z, H 4n (C(f); Z) = Z H i (C(f); Z)= 0 for i 0; 2n; 4n. H 2n (C(f); Z) H 4n (C(f); Z) α, β h(f) (h(f) Hopf ) α α = h(f)β (i) h(f ϕ) =deg(ϕ)h(f) ϕ : S 4n 1 S 4n 1 (ii) h(ψ f) =deg(ψ) 2 h(f) ψ : S 2n S 2n (iii) h(f 1 + f 2 ) = h(f 1 ) + h(f 2 ) f 1, f 2 : S 4n 1 S 2n 68. n = 1, 2, 4, 8, h(f) = 1 f : S 4n 1 S 2n h(f) = 3 n f : S 4n 1 S 2n 10

(6/28) 69. v 1 (v 2 c) = (v 1 v 2 ) c, v 1, v 2 H (X; R), c H (X; R). f : X X f (f v c) = v f c v H (X ; R), c H (X; R). 70 (Thom ). E Riemann m. D(E) := { x E x 1 }, S(E) := { x E x = 1 } S(E) ξ E = D(E)/S(E) Thom (i) B Thom E (ii) R k B B Thom (iii) S 1 Thom RP 2 71. Thom H i (B; R) = H i+m (ξ E ; R). 72. Wedge ξ E E = ξ E ξ E. 73. E B E B e(e) e(e ) = e(e E ). f : B B e(f E) = f (E) 74 (, p.178). n E B (i) Thom T (E) H n (E, E 0 ) ϕ : H n (B) = H 2n (B) ϕ (e(e)) = T (E) T (E). (ii) n 2e(E) = 0. 75 (Mathai-Quillen ). E B n B Hcomp(E) n n- 11

(7/5) 76. (i) R = Z (ii) B T M R = Z 77. Thom-Gysin 78. n G n (R ) E G n (R ) E E χ(e E) 79 (Bott-Tu, 12). p : E M M Euler χ(e) Poincaré 80. τ S n A S n S n E S n S n \ A H (E, E 0 ) = H (S n S n, S n S n \ ) = H (S n S n, A) n χ(τ) H n (S n ; Z) 2 τ 81. M m M M N T M 82. U M H m (M M; Z) N Thom a H k (M; Z) b H k (M; Z) a, b = ( 1) m+k2 U M, b (a [M]). 12

(7/12) 83 ( ). p : E B n F l n (E) Flag bundle q; F l n (E) B F l n (E) L 1,..., L n q (E) L 1 L n ( ) Leray-Hirsh q : H (B) H (F l n (E)) 84. X (1 ). (1) P(T X) X. (Hint ). x X, p : P(T X) X p 1 (x) ( P(T X), ). (2), l P(T X), X C - f l, f l x l., l U l,. (cf. ample) (3) U l1,..., U lm P(T X), l 1,..., l m, F = (f l1,..., f lm ) : X R m. (Hint: F, ) 85. Thom-Gysin U(n) H (U(n)) = Z[c 1,..., c n ]. 86. n E s w n (E) = 0, c n (E) = 0 w n (E) = 0 n E 87. E n E R E w 2n (E R ) c n (E) mod 2 c n (E) e(e R ) 13

(7/19) 88. Grothendieck Chern Chern 89. Chern 90. S P C n d α H 2 (P C n ; Z) S S Chern c(s) = (1 + a) n+1 (1 + da) 1. ( S N N = L d Whitney ) 91. n M R n+1, w i (T M) i w 1 (T M) i RP n R n+1 n = 2 r 1 n = 2 r 2 92. B E B n E E w 2 (E) = 0 93. C 2n C 2n P C 2n 1 P : C 2n 1 14