Morse ( ) 2014

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1 Morse ( ) 2014

2 1 1 Morse Morse Morse Smale Morse Morse Morse Morse Künneth Poincaré Euler Poincaré

3 1 1 Morse C 1.1 Morse M p T p M C F : M N p df p : T p M T F (p) N f : M R M C f p M df p : T p M T f(p) R = R 0 p f f(p) f a f f p a = f(p) M C f : M R p M (x 1,..., x n ) df p = i=1 f x i (p)dxi p dx 1 p,..., dx n p T p M (T p M ) p f f f (p) = = x1 x (p) = 0 n M f : M R p M X, Y T p M p X, Ỹ ( 2 f) p (X, Y ) = ( XỸ f)(p) ( 2 f) p (X, Y ) X, Y ( 2 f) p : T p M T p M R

4 p M M f : M R ( 2 f) p f p Hessian ( 2 f) p p f Morse

5 M C f : M R p M p (x 1,..., x n ) f f (p) = = x1 x (p) = 0 n X, Y T p M X = i=1 X i x i, Y = p i=1 Y i x i p Y p Ỹ = i=1 Ỹ i x i ( 2 f) p (X, Y ) = X(Ỹ f) = X i x i Ỹ j f i=1 p x j j=1 ( ) = X i Ỹ j f (p) xi x (p) + Y j 2 f j x i x (p) j i=1 = X i Y j 2 f x i x (p). j i,j=1 [ ] 2 f x i x (p) j x 1,..., p x n ( 2 f) p p p [ ] f 2 f x i x (p) j S 2 S 2 = {(x, y, z) R 3 x 2 + y 2 + z 2 = 1} f(x, y, z) = z S 2 f (0, 0, ±1) ε = ±1 f (0, 0, ε) (0, 0, ε) (x, y) (x, y) f f(x, y) = z = ε 1 x 2 y 2

6 f (0, 0) (0, 0, ε) f (x, y) f [ ] ε 0 0 ε ( 2 f) x ((x, y), (x, y)) = ε(x 2 + y 2 ) f S 2 Morse x x V n M V x x M (U, ϕ) ϕ(u V ) = ϕ(u) R d (R d = {(u 1,..., u d, 0,..., 0) R n u i R}) V M d ϕ ϕ(u) R d V d d V R n p R n Morse f p : V R ; x x p 2 V N(V ) E N(V ) = {(x, v) R n R n x V, v T x V } E : N(V ) R n ; (x, v) x + v f : M N x M df x : T x M T f(x) N x f f f f V N(V ) R n R n n (x, v) N(V ) E p = E(x, v) = x + v 2 f p x = 2, x 2 v, u i u j u i u j (i, j) 2 x u i u j.

7 x 0 V x 0 R n U C ϕ : U R n ϕ : U ϕ(u) ψ ψ (u 1,..., u n ) ψ ψ,..., ψ (1.1) u 1 u n ϕ(u) ϕ(u V ) ψ,..., ψ (1.2) u 1 u d U V U x (1.1) Gram- Schmidt (e 1 ) x,..., (e n ) x a 1 1(x) a 1 n(x) [ ψ [(e 1 ) x (e n ) x ] = (x) ψ ]. (x) 0... u 1 u n a n n(x) R n (e 1 ) x,..., (e n ) x (f 1 ) x,..., (f n ) x (x 0, v 0 ) N(V ) U R n (x 0, v 0 ) Φ : U R n R n R n ; (x, v) (ϕ(x), (f 1 ) x (v),..., (f n ) x (v)) Φ Φ((U R n ) N(V )) = Φ(U R n ) (R d R n d ) = Φ(U R n ) R n N(V ) R n R n n Φ (u 1,..., u d, t d+1,..., t n ) ( ψ(u 1,..., u d, 0,..., 0), k=d+1 t k (e k ) ψ(u1,...,u d,0,...,0) N(V ) (e 1 ) ψ,..., (e d ) ψ T ψ V (u 1,..., u d, t d+1,..., t n ) N(V ) E E(u 1,..., u d, t d+1,..., t n ) = ψ(u 1,..., u d, 0,..., 0) + k=d+1 ) t k (e k ) ψ(u1,...,u d,0,...,0)

8 (u 1,..., u d, t d+1,..., t n ) E = ψ e k + t k u i u i u i E t j = e j k=d Y 1,..., Y n R n R n X 1,..., X n [ X i, Y j ] de E u i, E t k R n ψ,..., ψ, e d+1,..., e n u 1 u d E u i, ψ E u j u i, e l = E t k, ψ E u j t k, e l [ E u i, ψ ] E u j u i, e l 0 1 n d d [ E, ψ ] u i u j E, ψ ψ = + u i u j u i k=d+1 ψ =, ψ + u i u j d + 1 k n ek, ψ + e k, u i u j v = k=d+1 x p = v 2 f p x = 2, x u i u j u i u j t k e k, 2 v, e k t k, ψ u i u j k=d+1 t k ek u i, ψ u j 2 ψ = 0 u i u j p = x + v 2 x u i u j. E = 2, ψ. u i u j (x, v) E

9 x V f p : V R p = x + v, v Tx V (x, v) N(V ) E p E f p Morse E : N(V ) R n R n p E f p Morse (Sard) f : M N N f 1.2 Morse Morse R 0 C f C g f(x) = f(0) + f (0)x + g(x)x 2, f (0) = 2g(0) p V f : V R p (U, ϕ) i ϕ(p) = 0, i f ϕ 1 (x 1,..., x n ) = f(p) x 2 j + x 2 j. j=1 j=i+1 V = R n p = 0 ( 2 f) 0 R n ( 2 f) 0 ( 2 f) 0 R n n n = R n = R R n 1 R n x R, y R n 1 (x, y) F (x, y) = f (x, y) x F (0, 0) = f (0, 0) = 0 x F x (0, 0) = 2 f (0, 0) 0 x2

10 R n 1 0 ϕ F (ϕ(y), y) = 0, ϕ(0) = 0

11 ϕ dϕ 0 = 0 R n (0, 0) Φ(x, y) = (x + ϕ(y), y) dφ (0,0) Φ (0, 0) f (f Φ)(0, y) = (ϕ(y), y) = F (ϕ(y), y) = 0 x x 2 (f Φ) (0,0) = 2 f (0,0) f f (0, y) = 0 x f f(x, y) = f y (x) f y y R n 1 f y C g y f y (x) = f y (0) + f y(0)x 2 + g y (x)x 2 f y(0) = f (0, y) = 0 x f y (x) = f y (0) + g y (x)x 2 0 f y (0) 0 f 0 (0) = 0 ( 2 f) 0 0 f f 0 (0) 0 0 f y (0) 0 n = 1 x 1 = ϵg y (x)x f = ϵx f y (0) = ϵx f(0, y) y f(0, y) R n n Morse i 1.2.4

12 n f i f n i 0 n f : M R g : N R f + g : M N R ; (x, y) f(x) + g(y) M N f + g (x, y) M N f + g x f y g f + g f g (x, y) M N f + g x f y g f + g Morse f Morse g Morse

13 (x, y) f +g f +g (x, y) f x g y R f : R R ; x cos(2πx) T 1 = R/Z f f df = 2π sin(2πx)dx R Z T 1 0, 1 2 cos Taylor f f Morse T 1 n T n = T 1 T f + + f : T n R Morse { (ϵ 1,..., ϵ n ) ϵ i = 0, 1 } 2 (ϵ 1,..., ϵ n ) #{i ϵ i = 0}

14 M X M I M c : I M dc dt (t) = X c(t) (t I) c X M X M X R X M X p M X c p : R M c p (0) = p ϕ t (p) = c p (t) ϕ t : M M t R ϕ t : M M (1) (3) (1) ϕ 0 = 1 M (2) s, t R ϕ t+s = ϕ t ϕ s (3) t R ϕ 1 t = ϕ t M t R ϕ t : M M {ϕ t } t R (1) (3) {ϕ t } t R M M {ϕ t } t R d dt ϕ t(p) = X p T p M t=0 M X X X {ϕ t } t R X {A λ } λ Λ X X = λ Λ A λ {A λ } λ Λ X Λ {A λ } λ Λ X X f supp(f) = {x X f(x) 0} f

15 M {U α } α A (1) (3) M {ϕ i } 1 i N (1) 0 ϕ i 1 (1 i N). (2) 1 i N α A supp(ϕ i ) U α (3) M N ϕ i = 1 i= {ϕ i } 1 i N {U α } α A (3) {supp(ϕ i )} 1 i N M 2.2 f R n f gradf ( f grad x f =,..., f ) x 1 x n R n, f df grad x f, Y = (df) x (Y ) (Y R n ) R n R n Riemann M p M T p M, p M C X, Y X, Y M C, M Riemann (M,, ) Riemann Riemann Riemann Euclid Riemann (M,, ) f gradf grad x f, Y x = (df) x (Y x ) (x M, Y T x M)

16 Riemann (1) (2) Riemann f gradf ϕ t d dt f(ϕ t(x)) = grad ϕt (x)f 2 < 0 f ϕ t (x) f : M R M Morse M X f (1) (df) x (X x ) 0 x f (2) f Morse X R n f 1

17 Morse i f(x 1,..., x n ) = f(p) x 2 j + j=1 j=i+1 gradf = ( 2x 1,..., 2x i, 2x i+1,..., 2x n ) x 2 j gradf = (2x 1,..., 2x i, 2x i+1,..., 2x n ) gradf dx(t) dt = grad x(t) f = (2x 1 (t),..., 2x i (t), 2x i+1 (t),..., 2x n (t)) x(t) = (x 1 (0)e 2t,..., x i (0)e 2t, x i+1 (0)e 2t,..., x n (0)e 2t ) gradf ϕ t ϕ t = diag(e 2t,..., e 2t, e 2t,..., e 2t ) ϕ t (x) = (e 2t x 1,..., e 2t x i, e 2t x i+1,..., e 2t x n ) Morse M f : M R Morse f M f f c 1,..., c r Morse (U 1, ϕ 1 ),..., (U r, ϕ r ) U 1,..., U r (U j, ϕ j ) (r + 1 j N) (U j ) 1 j N M c i (1 i r) U i 1 j N U j X j f ϕ 1 j R n ϕ j (U j ) R n f ϕ 1 j grad(f ϕ 1 j ) grad(f ϕ 1 j ) ϕ 1 j 1 X j (X j ) x = d(ϕ 1 j ) ϕj (x)(grad ϕj (x)(f ϕ 1 j )) (x U j )

18 df x ((X j ) x ) = df x (d(ϕ 1 j ) ϕj (x)(grad ϕj (x)(f ϕ 1 j ))) = d(f ϕ 1 j ) ϕj (x)(grad ϕj (x)(f ϕ 1 j )) = grad ϕj (x)(f ϕ 1 j ) 2 0. X j f U j (U j ) 1 j N (ψ j ) 1 j N { ψ j (x)x j (x) (x U j ) X j (x) = 0 (x / U j ) M X j X = N j=1 X j M X X f x M df x (X x ) = N df x (( X j ) x ) 0. j=1 j df x (( X j ) x ) = 0 j ψ j (x)x j (x) = 0 x f j ψ j (x) = 0 {ψ j } j x f X c i X X i f Morse R n f X f a M f ϕ t f a { } W s (a) = x M lim ϕ t(x) = a t + W u (a) = { x M } lim ϕ t(x) = a t

19 M Morse a W s (a) W u (a) V a Ind(a) dim W u (a) = codimw s (a) = Ind(a). M Morse f a Morse (U; x 1,..., x n ) M U R n i = Ind(a) f f(a) f f(a) = 0 f U R n x f(x 1,..., x n ) = i x 2 j + j=1 j=i+1 V = {(x 1,..., x i, 0,..., 0) x j R (1 j i)}, V + = {(0,..., 0, x i+1,..., x n ) x j R (i + 1 j n)} x 2 j x = x + x + (x V, x + V + ) f(x) = x 2 + x + 2 U(ϵ, η) U ϵ, η > 0 U(ϵ, η) = {x R n ϵ < f(x) < ϵ, x 2 x + 2 η(ϵ + η)} ± U = {x U f(x) = ±ϵ, x 2 η} ± U U(ϵ, η) 0 U = {x U x 2 x + 2 = η(ϵ + η)} 0 U U(ϵ, η) U(ϵ, η) = + U U 0 U V +, V W s (a) U = V + U, W u (a) U = V U f ϕ t Φ + : ( + U V + ) R M ; (x, t) ϕ t (x)

20 Φ + + U V + V + n i 1 W s (a) Φ + {a} n i W u (a) i M f M Morse f γ f c, d lim γ(t) = c, lim t γ(t) = d. t γ f γ γ f f Crit(f) Crit(f) x Crit(f) f Morse U(x) f(γ(t)) γ x Crit(f) U(x) lim f(γ(t)) = x t U(x) γ(r) = γ(i) I γ(t) I U(x) f(γ(t)) γ(t) U(x) γ(t) x Crit(f) γ U(x) t 0 t t 0 γ(t) U = U(x) x Crit(f) γ f X M U δ > 0 t t 0 f(γ(t)) f(γ(t 0 )) = t t 0 df x (X x ) δ (x M U) d dt f(γ(t))dt = δ(t t 0 ) t t 0 df γ(t) (γ (t))dt = t t 0 df γ(t) (X γ(t) )dt lim f(γ(t)) =. t f M d Crit(f) lim γ(t) = d t c lim γ(t) = c t

21 M f a M a = f 1 ((, a]) f M a, b [a, b] f M a M b (Reeb) M M Morse M Morse f f f f(m) = [0, 1] f Morse ϵ > 0 f 1 ([0, ϵ]) = M ϵ f 1 ([1 ϵ, 1]) n D n M 1 ϵ M ϵ M ϵ D n M 1 ϵ D n M = M 1 ϵ f 1 ([1 ϵ, 1]), M 1 ϵ f 1 ([1 ϵ, 1]) = M 1 ϵ = f 1 ([1 ϵ, 1]). M 1 ϵ f 1 ([1 ϵ, 1]) S n 1 S n 1 M 1 ϵ f 1 ([1 ϵ, 1]) S n 1 ϕ ϕ : S n 1 S n 1 M D n S n 1 ϕ M S n X, Y f, g : X Y (1) (2) F : X [0, 1] Y f g f, g (1) F (x, 0) = f(x) (x X), (2) F (x, 1) = g(x) (x X) f : X Y f : Y X f f 1 X, f f 1 Y f X Y

22 n D n {0} f : D n {0} ; x 0 f : {0} D n ; 0 0 f, f f f = 1 {0} F (x, t) = tx ((x, t) D n [0, 1]) F : D n [0, 1] {0} F (x, 0) = 0 = f f(x), F (x, 1) = x = 1 D n(x) f f 1 D n D n {0} M M ϕ : S k 1 M x S k 1 ϕ(x) M M B k M ϕ B k M B k

23 f M p f λ f(p) = c ϵ > 0 f 1 ([c ϵ, c + ϵ]) p f ϵ > 0 M c+ϵ M c ϵ B λ M c ϵ B λ 2.3 Smale M X Y x X Y T x X + T x Y = T x M X Y X Y M X Y X Y M T x (X Y ) = T x X T x Y (x X Y ), dim(x Y ) = dim X + dim Y dim M Morse f Smale M Morse f (1) a W u (a) W s (a) (2) a b f(a) f(b) W u (a) W s (b) = W u (a) W s (b) M Morse f a f(w s (a)) [f(a), ), f(w u (a)) (, f(a)] f(w s (a) {a}) (f(a), ), f(w u (a) {a}) (, f(a))

24 (W s (a) {a}) (W u (a) {a}) = W s (a) W u (a) = {a} a T a (W s (a)) + T a (W u (a)) = V + + V = T a M W s (a) W u (a) f(w u (a) {a}) (, f(a)), f(w s (b) {b}) (f(b), ) (W u (a) {a}) (W s (b) {b}) = a b W u (a) W s (b) = W u (a) W s (b)

25 f M Morse f X Smale f a, b W u (a) W s (b) M(a, b) = W u (a) W s (b) (1) f(a) > f(b). (2) M(a, b) M { } M(a, b) = x M lim ϕ t(x) = a, lim ϕ t (x) = b, t t dim(m(a, b)) = Ind(a) Ind(b) Ind(a) > Ind(b) (3) X M ϕ t M(a, b) (4) ϕ t M(a, b) L(a, b) L(a, b) a b X dim L(a, b) = Ind(a) Ind(b) 1. (5) f(a) > α > f(b) f α M(a, b) f 1 (α) M L(a, b) (1) (2) (2) W u (a) W s (b) { } M(a, b) = x M lim ϕ t(x) = a, lim ϕ t (x) = b t t Smale W u (a) W s (b) M(a, b) = W u (a) W s (b) M dim W u (a) + dim W s (b) = n + dim(w u (a) W s (b)) dim W u (a) = Ind(a) dim W s (b) = n Ind(b) Ind(a) Ind(b) = dim(w u (a) W s (b)) = dim(m(a, b)) dim(m(a, b)) 1 Ind(a) > Ind(b) Morse f Morse

26 f Morse U M U M U df(x) < 0 ϵ 0 > 0 M U df(x) < ϵ 0 M h (1) f Morse (2) M U dh(x) < 1 2 ϵ 0 (3) f c c f(c) + h(c) f(c ) + h(c ). f + h Morse (1) f + h f (2) X f + h (3) f + h

27 (Smale) M f M Morse X f X X Smale

28 26 3 Morse M f M Morse X f Smale 3.1 Morse Morse f a b L(a, b) = L(a, c 1 ) L(c 1, c 2 ) L(c q 1, b) c i Crit(f) f(c i ) f(c i+1 ) L(c i, c i+1 ) = f(c i ) > f(c i+1 ) Ind(c i ) > Ind(c i+1 ) L(a, b) Ind(a) = Ind(b) + 1 L(a, b) Ind(a) = Ind(b) + 2 L(a, b) S 1 [0, 1] Z 2Z Z 2 = Z/2Z f k Crit k (f) Crit k (f) Z 2 C k (F ) f X X : C k (f) C k 1 (f) a Crit k (f) X (a) b Crit k 1 (f) L(a, b) #L(a, b) = #L(a, b) Z 2 n X (a, b) X (a) = n X (a, b)b X (a) X X = 0 b Crit k 1 (f) (C k (f), X ) M Morse X Morse C k+1 (f) X X = 0 C k (f) im X ker X H k (f, X) = ker X /im X H k (f, X) (C k (f), X )

29 Morse n S n = {x R n+1 x = 1} f(x 1,..., x n+1 ) = x n+1 ((x 1,..., x n+1 ) S n ) f n = (0,..., 0, 1), s = (0,..., 0, 1) f S n Morse Ind(n) = n, Ind(s) = 0 C n (f) = Z 2 n, C 0 (f) = Z 2 s, C k (f) = {0} (1 k n 1) f n 2 0 H n = Z 2 n = Z 2, H 0 = Z 2 s = Z 2, H k = {0} (1 k n 1). n = 1 L(n, s) : C 1 (f) C 0 (f) 0 0 H 1 = Z 2 n = Z 2, H 0 = Z 2 s = Z RP n n S n π : S n RP n S n S n ; x x R n+1 (x 0, x 1,..., x n ) R n+1 f f(x) = kx 2 k (x R n+1 ) k=0 f x x RP n f S n = {x R n+1 x = 1} U + i = {x S n x i > 0}, U i = {x S n x i < 0} (x 0,..., ˆx i,..., x n )

30 U ± i U ± i S n U ± i f f(x) = kx 2 k = ( kx 2 k + i 1 ) x 2 k = i + (k i)x 2 k k=0 k i k i k i f U ± i i p ± i = (0,..., 0, ±1, 0,..., 0) f p ± i Ind(p ± i ) = #{k 0 k n, k i < 0} = i f S n Morse U ± i ( i ) 2 ( k ) 2 kxk + ixk f(x) = i k<i k>i y k = i kx k (k < i), y k = k ix k (k > i) f Morse Morse f 1 gradf = 2y y i 1 2y i y n. y 0 y i 1 y i+1 y n ϕ t x k ϕ t (y) = (e 2t y 0,..., e 2t y i 1, e 2t y i+1,,e 2t y n ) ϕ t (x) = (e 2t x 0,..., e 2t x i 1, e 2t x i+1,,e 2t x n ) f X p ± i V ± i U ± i f Morse 1 W s (p ± i ) V ± i = {(0,..., 0, x i+1,..., x n ) V ± i } = {(0,..., 0, x i, x i+1,..., x n ) S n } V ± i, W u (p ± i ) V ± i = {(x 0,..., x i 1, 0,..., 0) V ± i } = {(x 0,..., x i 1, x i, 0,..., 0) S n } V ± i. W u (p ± 0 ) = {p ± 0 }, W s (p ± n ) = {p ± n } ( )

31 i n C i ( f) = Z 2 π(p + i ) (π(p + i )) = π(p+ i 1 ) + π(p i 1 ) = 2π(p+ i 1 ) = 0. H i = C i ( f) = Z 2 π(p + i ) = Z C n+1 1 n CP n C n+1 {0} z z 1 Cz CP n π : C n+1 {0} CP n ; z Cz π C n+1 {0} CP n C n+1 C n+1 = {(z 0,..., z n ) z i C} 0 i n U i CP n U i U i = {π(z) z C n+1, z i 0} CP n = n i=0 U i w k (π(z)) = z k z i (0 k n) w k (w 0,..., ŵ i,..., w n ) U i n CP n w i = 1 C n+1 {0} f f(z) = n k=0 k z k 2 n k=0 z k 2 (z C n+1 {0}) α C {0} f(αz) = f(z) C n+1 {0} f CP n f f(π(z)) = f(z)

32 f U i f ( ) f(π(z)) = f(π(w)) = n k=0 k w k 2 n k=0 w k 2 = i + k i k w k k i w k 2 U i f w = 0 CP n p i = π(0,..., 0, i 1, 0,..., 0) p i f ( ( ) = i + ) k w k 2 1 ( ) 2 w k 2 + w k 2 k i k i k i = i + k i k w k 2 i k i w k 2 + (4 ) = i + k i (k i) w k 2 + (4 ) = i k<i (i k) w k 2 + k>i (k i) w k 2 + (4 ). p i f Ind(p i ) = 2#{k 0 k n, k < i} = 2i f(p i ) = i C 2i (f) = Z 2 p i, C j (f) = {0} (j ). f 0 Morse (C i (f), ) H 2i = C 2i (f) = Z 2 p i = Z2, H j = {0} (j ). S 2 S 2 Morse Morse 8 6 B720

33 31 4 Morse 4.1 M Morse H (f, X) M HM (M; Z 2 ) M 2 Morse Künneth Künneth Morse U, V Z 2 u i v j U V U V = Z 2 u i v j i,j U V u i v j U V Z 2 dim U V = dim U dim V (u i, v j ) u i v j Z 2 U V U V (u, v) U V u v Z 2 Z 2 ϕ : U U, ψ : V V Z 2 ϕ ψ : U V U V (ϕ ψ)(u v) = ϕ(u) ψ(v) (u U, v V )

34 C = (C, C ) C i Z 2 Z 2 i C C i C i 1 i C i+1 C = 0 ker i C im i+1 C H i (C) = ker C i /im C i+1 H (C) = (H i (C)) C C = (C, C ) D = (D, D ) Z 2 (C D) k = C i D j i+j=k = ( C i j D C D k i+j=k ) : (C D)k (C D) k 1 Z 2 ( C i D j ) (Ci D j ) C i 1 D j + C i D j 1 (C D) k 1 (C D, d C D ) c C i d D j (d C D ) 2 (c d) = d C D ( C i (c) d + c D j (d)) = C i 1 C i (c) d + 2 C i (c) D j (d) + c D j 1 D j (d) = Z 2 C, D H k (C D) = H i (C) H j (D) i+j=k

35 l(d) = max{j D j {0}} min{j D j {0}} + 1 l(d) = 1 j 0 D j0 {0} D j {0} H k (C D) = H k j0 (C) D j0 = H k j0 (C) H j0 (D) l(d) = 1 l(d) = 2 j 0 D j0 {0} D j0 1 {0} D j {0} D = 0 D = 0 D = 0 C D = C 1 H k (C D) = H k j0 (C) D j0 H k j0 +1(C) D j0 1 = H k j0 (C) H j0 (D) H k j0 +1(C) H j0 1(D) D j 0 : D j0 D j0 1 H (D) = 0 H (C D) = 0 C D i+j 0 : C i D j0 C i+1 D j0 1 C i 1 D j0 C i D j0 1 ker C D i+j 0 C D i+j 0 +1 : C i+1 D j0 C i+2 D j0 1 C i D j0 C i+1 D j0 1 im C D i+j 0 +1 ker C D i+j 0 im C D ker C D i+j 0 i+j 0 +1 C i D j0 C i+1 D j0 1 ( C D i+j 0 x i a y j 0 a, ) u i+1 b v j 0 1 b = 0 a b a x i a D j 0 (y j 0 a ) + b a C i (x i a) y j 0 a = 0 i+1(u C i+1 b ) v j 0 1 b = 0 1 ( D j 0 ) 1 a x i a y j 0 a + b i+1(u C i+1 b ) ( j D 0 ) 1 (v j 0 1 b ) = 0

36 a x i a y j 0 a = b i+1(u C i+1 b ) ( j D 0 ) 1 (v j 0 1 b ). C D i+j 0 +1 ( b u i+1 b ( D j 0 ) 1 (v j 0 1 b ), 0 ) = ( a x i a y j 0 a, b u i+1 b v j 0 1 b ). ker C D i+j 0 im C D i+j 0 +1 H (C D) = 0 l(d) = 2 D D j0 D j0 1 D j0 = ker D j 0 D j 0, D j0 1 = im D j 0 D j D j0 D j0 1 0 E : 0 ker D j 0 0 D j 0 1 0, F : 0 D j 0 = im D j 0 0 D = E F C D = C (E F ) = C E C F C D C E, C F C D D k D k + 1 D 0 D k+1 D k D 1 0 D k+1 D k D k+1 = ker D k+1, D k = im D k D 0 ker 0 0 D k+1 = im 0 0 D k D k 1 D 1 0 k D 4.1.1

37 M N Morse f g X Y Smale f + g M N Morse (X, Y ) Smale f + g Crit(f + g) = Crit(f) Crit(g) (a, a ) Crit(f + g) = Crit(f) Crit(g) Ind(a, a ) = Ind(a) + Ind(a ) Crit k (f + g) = i+j=k Crit i (f) Crit j (g) (b, b ) Crit k 1 (f + g) (a, a ) (b, b ) (X, Y ) (X, Y ) X Y L (X,Y ) ((a, a ), (b, b )) = L X (a, b) L Y (a, b ) a b a b L (X,Y ) ((a, a ), (b, b )) L (X,Y ) ((a, a ), (b, b )) L X (a, b) L Y (a, b ) Ind(a) Ind(b) + 1, Ind(a ) Ind(b ) + 1 Ind(a, a ) = Ind(a) + Ind(a ) Ind(b) + Ind(b ) + 2 = Ind(b, b ) + 2 Ind(a, a ) = k, Ind(b, b ) = k 1 a b a b L (X,Y ) ((a, a ), (b, b )) = (a, a ) (b, b ) (X, Y ) a = b a = b L (X,Y ) ((a, a ), (b, b )) = { {a} L Y (a, b ) (a = b) L Y (a, b) {a } (a = b ) n Y (a, b ) (a = b) n (X,Y ) ((a, a ), (b, b )) = n X (a, b) (a = b ) 0 ( ) Φ : C i (f) C j (g) C k (f + g) i+j=k

38 Φ(a a ) = (a, a ) Φ Z Φ (C (f) C (g), X Y ) (C (f + g), (X,Y ) ) a Crit i (f) a Crit j (g) Φ ( X Y )(a a ) = Φ( X (a) a + a Y (a )) = Φ n X (a, b)b a + a n Y (a, b )b = b Crit i 1 (f) b Crit i 1 (f) n X (a, b)(b, a ) + b Crit j 1 (g) b Crit j 1 (g) n Y (a, b )(a, b ). L (X,Y ) ((a, a ), (b, b )) n (X,Y ) ((a, a ), (b, b )) (X,Y ) Φ(a a ) = n (X,Y ) ((a, a ), (b, b ))(b, b ) = (b,b ) Crit i+j 1 (f+g) b Crit i 1 (f) n X (a, b)(b, a ) + b Crit j 1 (g) Φ ( X Y )(a a ) = (X,Y ) Φ(a a ) Φ n Y (a, b )(a, b ) (Künneth ) M N HM k (M N; Z 2 ) HM i (M; Z 2 ) HM j (M; Z 2 ). i+j=k

39 Poincaré n Morse f k f n k X f X f a Crit k (f) f a a Crit n k ( f) Z 2 V V C k (f) C n k ( f) X : C k (f) C k 1 (f) t X : C k 1 (f) C k (f) t X (α)(c) = α( X (c)) (α C k 1 (f), c C k (f)) t X : C n k+1 ( f) C n k ( f) (Poincaré ) n M HM n k (M; Z 2 ) HM k (M; Z 2 ) 4.2 Euler Poincaré Morse n M Euler χ(m) χ(m) = ( 1) k dim HM k (M; Z 2 ). k=0 χ(m) = ( 1) k dim HM k (M; Z 2 ) = = k=0 ( 1) k (dim ker k dim im k+1 ) k=0 ( 1) k (dim ker k + dim im k ) = k=0 ( 1) k dim HM k (M; Z 2 ) (Morse ) M Morse f c k (f) = #Crit k (f), β k = dim HM k (M; Z 2 ) k 0 c k (f) β k n = dim M #Crit(f) β k. k=0 k=0

40 n M Poincaré P M (t) = β k t k k=0 P M (1) = β k, k=0 P M ( 1) = χ(m) (1) Poincaré (2) M, N P M N (t) = P M (t)p N (t) B720

2 1 κ c(t) = (x(t), y(t)) ( ) det(c (t), c x (t)) = det (t) x (t) y (t) y = x (t)y (t) x (t)y (t), (t) c (t) = (x (t)) 2 + (y (t)) 2. c (t) =

2 1 κ c(t) = (x(t), y(t)) ( ) det(c (t), c x (t)) = det (t) x (t) y (t) y = x (t)y (t) x (t)y (t), (t) c (t) = (x (t)) 2 + (y (t)) 2. c (t) = 1 1 1.1 I R 1.1.1 c : I R 2 (i) c C (ii) t I c (t) (0, 0) c (t) c(i) c c(t) 1.1.2 (1) (2) (3) (1) r > 0 c : R R 2 : t (r cos t, r sin t) (2) C f : I R c : I R 2 : t (t, f(t)) (3) y = x c : R R 2 : t (t,