M ω f ω = df ω = i ω idx i f x i = ω i, i = 1,..., n f ω i f 2 f 2 f x i x j x j x i = ω i x j = ω j x i, 1 i, j n (3) (3) ω 1.4. R 2 ω(x, y) = a(x, y

1 1.1 M n p M T p M Tp M p (x 1,..., x n ) x 1,..., x n T p M dx 1,..., dx n Tp M dx i dx i ( ) = δj i x j Tp M Tp M i a idx i 1.1. M x M ω(x) Tx M ω(x) = n ω i (x)dx i i=1 ω i C r ω M C r C ω( x i ) C r 1. x i y j x i x i = j y j dx i = j x i y j dy j (1) 1.1 ( ). M f df x : T x M T f(x) R = R Tx M df x = i f x i (x)dx i x df x x 1,..., x n dx 1,..., dx n dx 1,..., dx n 1.2 ( ). R 2 (r, θ) Φ : (0, ) R (r, θ) (r cos θ, r sin θ) R 2 \ {0} (2) Φ 1 R 2 \{0} (r, θ) dr, dθ dr, dθ R 2 \{0} (r 1, θ 1 ), (r 2, θ 2 ) r 1 = r 2, θ 1 θ 2 + 2kπ θ (1) dθ, dr 1.3 ( ). T n = R n /Z n π : R n T n R n x x + c R n 1.2 dx 1,..., dx n T 1

M ω f ω = df ω = i ω idx i f x i = ω i, i = 1,..., n f ω i f 2 f 2 f x i x j x j x i = ω i x j = ω j x i, 1 i, j n (3) (3) ω 1.4. R 2 ω(x, y) = a(x, y)dx + b(x, y)dy (3) a y = b x o = (0, 0), p = (x, y) γ F γ (x, y) = a(x, y)dx + b(x, y)dy γ ξ = (x, 0), η = (0, y) γ ξ γ 1 η γ 2 R F γ1 (x, y) F γ2 (x, y) = R a(x, y)dx + b(x, y)dy = R ( a y b x )dxdy = 0 F (x, y) = F γ1 (x, y) = F γ2 (x, y) F γ 1 y (x, y) = b(x, y) γ F 2 x (x, y) = a(x, y) df = ω R 2 (3) R n, B n 1.5. 1.2 Ω = R 2 \ {0} dr Ω r(x, y) = x 2 + y 2 dθ dθ f Ω (2) Φ F (r, θ) = f Φ(r, θ) F (r, θ + 2π) = F (r, θ) Φ df = dθ F θ (r, θ) 1, F r 0 F (r, θ) = θ + c F dθ (3) 1.3 dx 1,..., dx n T n dx i (3) 2

1.5 dθ (r, θ) x θ 1.4 F {0} 2π dθ (3) 1.2 p (3) (x 1,..., x n ), (y 1,..., y n ) 1-formω ω(x) = ω i (x)dx i = η j (y)dy j y 1-form ω i = η k k ω i x j ω j x i = x j (η k y k x i ) x i (η k y k x j ) = η k y k η k y k + η k ( 2 y k 2 y k ) x j x i x i x j x j x i x i x j = η k y l y k η k y l y k = ( η k η l ) y l y k (k, l ) y l x j x i y l x i x j y l y k x j x i = 1 ( η k η l )( y k y l y l y k ) (4) 2 y l y k x i x j x i x j 1 k,l n (3) Γ ij = ω i x j ω j x i 2 Γ ij V = T x M n p { }} { V p f : V V R i v 1,..., ˇv i,..., v n V v i f(v 1,..., v n ) R f : V V R f(v, w) = f(w, v) V 1.2. V p f p σ f(v 1,..., v p ) = sgn(σ)f(v σ(1),..., v σ(p) ) f V p p-form V p-form p V n Σ n Σ n n! ω p V, η q V ω η p+q V ω η(v 1,..., v p+q ) = 1 p!q! σ Σ p+q sgn(σ)ω(v σ(1),..., v σ(p) )η(v σ(p+1),..., v σ(p+q) ) x i 3

ω η(v τ(1),..., v τ(p+q) ) = 1 p!q! ω η τ Σ p+q σ Σ p+q sgn(σ)ω(v τσ(1),..., v τσ(p) )η(v τσ(p+1),..., v τσ(p+q) ) Σ p+q σ τσ σ = τσ ω η(v τ(1),..., v τ(p+q) ) = 1 p!q! σ Σ p+q sgn(τ 1 σ )ω(v σ (1),..., v σ (p))η(v σ (p+1),..., v σ (p+q)) = sgn(τ)ω η(v 1,..., v p+q ) ω η 1.1. ω 1, ω 2, ω 3 V p, q, r form (ω 1 ω 2 ) ω 3 = ω 1 (ω 2 ω 3 ) (5) ω 1 ω 2 = ( 1) pq ω 2 ω 1 (6) Proof. (5) 1 p!q!(p + q)!r! σ Σ p+q+r τ Σ p+q sgn(σ) sgn(τ)ω 1 (v στ(1),..., v στ(p) ) ω 2 (v στ(p+1),..., v στ(p+q) )ω 3 (v σ(p+q+1),..., v σ(p+q+r) ) τ Σ p+q {p + q + 1,..., p + q + r} p + q + r Σ p+q Σ p+q+r τ σ = στ 1 p!q!(p + q)!r! τ Σ p+q σ Σ p+q+r sgn(σ )ω 1 (v σ (1),..., v σ (p)) ω 2 (v σ (p+1),..., v σ (p+q))ω 3 (v σ (p+q+1),..., v σ (p+q+r)) Σ τ τ Σ p+q /p!q!(p+ q)!r! = 1/p!q!r! (5) 1 p!q!r! σ Σ p+q+r sgn(σ)ω 1 (v σ(1),..., v σ(p) )ω 2 (v σ(p+1),..., v σ(p+q) )ω 3 (v σ(p+q+1),..., v σ(p+q+r) ) (6) τ(i) = i+q, 1 i p, τ(p+j) = j, 1 j q τ sgn τ = ( 1) pq 1. 1.1 4

V v 1,..., v n v 1,..., v n V I = (i 1,..., i p ) 1-form p v I = v i 1 v i p (6) v i v i = 0 I J = (j 1,..., j p ) v I (v J ) := v I sgn σ if j α = i σα (v j1,..., v jp ) = (7) 0 otherwise 1.2. v 1,..., v n {v I } 1 i1 <...i p n = {v i 1 v ip } 1 i1 <...i p n p V p > n p V = {0} p V nc p Proof. i 1 < < i p I I I a Iv I = 0 v j1,..., v jp (7) a J = 0 ω p V η = I ω(v I)v I J η(v J ) = ω(v J ) ω = η ω v I V v i = A j i w j ω p V ω = I a Iv I = J b Jw J (v i1,..., v ip ) (7) a I = J K b J A k 1 i 1... A k p i p w J (w K ) = sgn(σ)a jσ(1) J i 1 σ Σ p... A j σ(p) i p 1.6. p = 2 ω = i<j a ijv i v j = k<l b klw k w l b J a ij = k<l(a k i A l j A l ia k j )b kl (8) p = n (det A j i )v1 v n = w 1 w n 1.3 ( ). M x p Tx M ω x ω x = I ω I(x)dx I ω I x ω x M p p-form 1.7. M 1-formω(x) = i ω i(x)dx i dω = i,j ω i dx j dx i = ( ω j ω i )dx i dx j x j x i<j i x j 5

2-form dω M 2-form (4) ω i x j ω j x i = 1 2 = 1 k,l n 1 k<l n ( η k η l )( y k y l y l y k ) y l y k x i x j x i x j ( η k η l )( y k y l y l y k ) y l y k x i x j x i x j T x M x i = y j x i y j dω (8) 2-form ω 1-form α dα = ω p 2. 2-form ω 1-form α ω = dα 1.3 Ω p (M) M C p-form p = 0 C d 0 = d : Ω 0 (M) Ω 1 (M) d 1 = d : Ω 1 (M) Ω 2 (M) d 1 d 0 = 0 Im d 0 ker d 1 Ω p (M)... d p 2 Ω p 1 (M) d p 1 Ω p (M) d p Ω p+1 (M) d p+1... {d p } p d p d p 1 = 0 p d p d p-formω ω = I ω I(x)dx I d 1 d p ω = I dω I dx I = ω I dx j dx i 1 dx i p (9) x j I j d 1 1.3. ω, η p, q-form (1) d(ω η) = dω η + ( 1) p ω dη (2) d 2 ω = d p+1 d p ω = 0 6

Proof. ω = I ω Idx I, η = J η Jdx J ω η = I,J ω Iη J dx I dx J d (6) d(ω η) = d(ω Iη J )dx I dx J = ( ω I η J η J + ω I )dx k dx I dx J I,J I,J,k x k x k = { ωi dx k dx I η J dx J + ( 1) p ω I dx I η } J dx k dx J I,J,k x k x k = dω η + ( 1) p ω dη (3) f d 1 d 0 f = 0 1.3(1) d 2 ω = I d2 ω I dx I = 0 d p d p p d p : Ω p (M) Ω p+1 (M) (A) d 0 (B) d 1.3 d d 1.4. (1) d 1.3(1) d x M ω 0 dω(x) = 0 (2) (A) (B) d (9) Proof. (1) x U ω 0 φ φ(x) = 1 U bump function 0 d(φω) = dφ ω + φdω dω(x) = 0 d ω = I ω Idx I d(ω) = { dωi dx I + ω d(dx i 1 dx i p ) } I = dω + { } ω d 2 x i 1 dx ip ± dx i 1 d 2 x i 2 dx ip... = dω I (2) (A) (B) d (9) d 1.4. M {U α } α A M {χ α } α A (1) M χ α (x) 0 M \ U α χ α 0 (2) α A χ α 1 7

1.5. M {U α } α χ α Proof. M x U x φ x B 1 R n V x = φ 1 x (B 1 ) 2 bump functionθ x V x θ x 1 U 0 {V x } x M M {V α } α bump function θ α χ α (x) = θ α(x) α θ α(x) V χ α 1 dω = α d α(χ α ω) d α U α (9) U α d α (χ α ω) = 0 (p + 1)-form 1.6. d (A) (B) Proof. (A) 1.3 1.3 d α d(ω η) = α d α (χ α ω η) = α {d α (χ α ω) η + ( 1) p ω χ α d α η} = dω η + ( 1) p ω ( α χ α d α η) χ α d α η = d α (χ α η) d 0 χ α η U α M d(ω η) = dω η + ( 1) p ω dη α dχ α ω η α χ α 1 d 2 = 0 d 2 f = d(df) = 0 d(df) = α d α (χ α df) = α {dχ α df + χ α d 2 αf} = 0 (9) 1-formω dω(x, Y ) = X(ω(Y )) Y (ω(x)) ω([x, Y ]) 3. X 1,..., X p+1 M ω p-form F (X 1,..., X p+1 ) = p+1 ( 1) i+1 X i {ω(x 1,..., ˆX i,..., X p+1 )} i=1 + i<j ( 1) i+j ω([x i, X j ], X 1,..., ˆX i,..., ˆX j,..., X p+1 ) 8

[X i, X j ] X i { } F (p + 1)-form dω (1) (X 1,..., X p+1 ) F (X 1,..., X p+1 ) F I := F ( x i1,..., x ip+1 ) dω = I F Idx I (2) f F (fx 1,..., X p+1 ) = ff (X 1,..., X p+1 ) F (X 1,..., X p+1 ) F (X 1,..., X p+1 ) x M X 1 (x),..., X p+1 (x) T x M M m, N n F : M N N p-formω M p-formf ω F ω x (X 1,..., X p ) = ω F (x) (df x (X 1 ),..., df x (X p )) F ω ω F 1.8 ( ). M N ι T x M T x N N p-form ω T x M M p-form ι ω N (x 1,..., x n ) U φ(m U) = {(x 1,..., x n ); x m+1 = = x n = 0} ω ω = I ω Idx I M (x 1,..., x m ) ι ω = I {1,...,m} ω I M dx I dx m+1,..., dx n 1.7. M, N, F ω, η N (1) G : N X X α F G α = (G F ) α (2) F (ω η) = F ω F η (3) df ω = F dω Proof. (1) (2) (3) N f(x) (y 1,..., y n ) ω = I ω Idy I (2) F ω = I F ω I F dy i 1 F dy i p N f F df = df df = d(f F ) = df f 1.3(1) d 2 y i = 0 df ω = I F dω I F dy i 1 F dy ip = F dω 9

d p : Ω p (M) Ω p+1 (M) Z p (M) = ker d p, B p (M) = d p 1 Ω p 1 (M) Z p (M) B p (M) d 2 = 0 Z p B p Ω p (M) H p DR (M) = Zp (M)/B p (M) H p DR (M) M p 1.8. (1) M A HDR 0 (M) = α A R M HDR 0 (M) = R (2) F : M N F F : H p DR (N) Hp DR (M) F - Proof. (1) HDR 0 (M) = Z0 (M) Z 0 (M) (2) 1.7(3) F N M F : Z p (N) Z p (M) F : H p DR (N) Hp DR (M) 1.4 n M n-formω (y 1,..., y n ) f ω x = f(y)dy 1 dy n (x 1,..., x n ) ω x = f(y)dy 1 dy n = f(y) det( yi x j )dx1 dx n (10) y = Φ(x) ΦΩ f(y)dy 1... dy n = Ω f(φ(x)) det( Φi x j ) dx1... dx n n-form n-form 1.5. M {(U α, φ α )} α α, β y = φ β φ 1 α (x) det( yi ) > 0 x j M M M M 1.9. (1) n M M n-form M (2) M 10

Proof. M (U α, φ α ) (x 1,..., x n ) ω α = dx 1 dx n ω α 1 χ α ω = α χ αω α (10) ω = fdx 1 dx n f n-formω M (x 1,..., x n ) ω(x) = f(x)dx 1 dx n f ( x 1, x 2..., x n ) ω f (10) M n-formω f M n-formω 1, ω 2 S = ω 1 /ω 2 S ω 1 = Sω 2 M S S ω 1, ω 2 4. S n R n+1 A S n f A : S n S n RP n S n x S n f E (x) = x S n π : S n RP n (1) R n+1 n-formv n+1 V = ( 1) i 1 x i dx 1... dxi ˆ dx n+1 i=1 V S n n-form S n (2) A V = det AV f A (3) RP n n-formω f E π ω = π ω n RP n 1.6 ( ). M n Ω M Ω x M (U, φ) φ(ω U) = {(x 1,..., x n ) φ(u); x n 0} H n = {(x 1,..., x n ) R n ; x n 0} 11

Ω M Ω H n R n Ω Ω φ 2 φ 1 1 ( H φ 1(U 1 )) H (11) Ω M Ω (n 1) 1. 1.10. M Ω M Ω Proof. M Ω x Ω (U, φ), (V, ψ) F = ψ φ 1 Ω φ(x) = ψ(x) = 0 x n 0 F n (x n ) 0 F n x n (0) 0 (11) F n x i (0) = 0, i < n F 1 F x 1...... 1 x n J 1 (F ) =.... F (n 1) F x 1...... (n 1) x n 0 0 0 F n x n (n 1) Ω J 2 F 0 < det J 1 (F ) = det J n 2 x n (0) det J 2 > 0 n M (n 1) Σ Ω Σ M 1.9 ( ). S 2 S 2 S 1 T 2 S 1 1.10 1.10 Σ M 12

1.10. RP 2 RP 2 S 1 RP 3 RP 2 RP 3 1.10 RP 3 \ RP 2 3 RP 2 RP 3 1.11. Ω M {U α } α M Ω (1) α V α U α V α U α {V α } α (2) {U α } Ω {χ α } α χ α (a) χ α K U α (b) x Ω α χ α(x) 1 Proof. bump function U R n K U K f > 0 R n f 0 U 1.4 x Ω x U α α x W x U α U α W x W 1,..., W k W i α W i U α V α = Wi U α W i 1.5 M n Ω M Ω n-formω M Ω {(U α, φ α )} 1.11 1 χ α χ α ω U α U α ω = f α dx 1 dx n I (Uα,φ α )(χ α ω) := χ α (x)f α (x)dx H 1... dx n (12) + ω Ω ω := I Uα (χ α ω) Ω α 1.12. (12) U α, φ α, χ α 13

Proof. {(V β, ψ β )} β ν β y 1,..., y n V β ω = g β dy 1 dy n {(U α V β, φ α )} α,β, {(U α V β, ψ α )} α,β χ α ν β U α V β I (Uα V β,ψ β )(χ α ν β ω) = χ α (y)ν β (y)g β (y)dy ψ α (U α V β ) H 1... dy n + = χ α (x)ν β (x)g β (y) det( φα(uα Vβ) H yi + x j ) dx1... dx n = χ α (x)ν β (x)f α (x)dx φ α (U α V β ) H 1... dx n = I (Uα V β,φ α )(χ α ν β ω) + I Uα (χ α ω) = I (Uα,φ α )(χ α ν β ω) α α β = I (Uα Uβ,ψ β )(χ α ν β ω) = I Vβ (ν β ω) β α β Ω (n 1)- η ω = dη η U α η = n i=1 η i(x)dx 1... dx i dx n dη = n i=1 ( 1) i 1 η i x i (x)dx1 dx n n I Uα = H χ α ( 1) i 1 η i x i (x)dx1... dx n i=1 dx i i < n φ α (U α ) H χ η i α x i (x)dx1... dx n = H η χ α i x i dx1... dx n i = n H φ α(u α) H χ η n α x n (x)dx1... dx n = H η nχ α dx 1... dx n 1 H η χ α n x n dx1... dx n I α = ( 1) n H η nχ α dx 1... dx n 1 i ( 1) H i 1 η χ α i x i dx1... dx n dχ α η dχ α η = i,j χ α x j dxj η i dx 1... dx i dx n = i ( 1) i 1 χ α x i η idx 1 dx n 1.12 U α dχ α η 14

1.13 ( ). n M Ω Ω (n 1)-form η dη = Ω η Ω Ω η Ω η Ω M (x 1,..., x n ) Ω (x 1,..., x n 1 ) n Proof. dη = I α = } {( 1) n Ω α α H η nχ α dx 1... dx n 1 H dχ α η η Ω η n dx 1... dx n 1 α χ α 1 1. M n (1) M (n 1)-form η M dη = 0 (2) (n 1)-form η dη = 0 M Proof. 1.13 Ω p cpt (M) M p-form d(ωp cpt (M)) Ωp+1 cpt (M) H p cpt (M) = ker{d : Ω p(m) Ω p+1 (M)}/d(Ω p 1 cpt (M)) M n I M : H n (M) [ω] M ω R I(dη) = 0 well-defined 1.9 M M n-formvol I M (vol) > 0 I M H n (M) 0 M I M : Hcpt(M) n R n-form I M 0, Hcpt(M) n 0 M I M (n 1) Σ n 1 M n [η] H n 1 (M) [η Σ ] H n 1 (Σ) Σ Ω Ω = Σ I Σ ([η]) = Ω dη Σ = 0 15

[η] = 0 f : Σ M Ω = Σ f f : Ω M I Σ ([f η]) = d f η = f dη = 0 Ω Ω Σ [f η] = 0 [η Σ ] 0 2 2.1 2.1. M, N f 0, f 1 : M N F : M [0, 1] N F (x, i) = f i (x), x M f 0, f 1 F f 0, f 1 f 0 f 1 f : M N g : N M g f id M, f g id N f M, N 1. F : M [0, 1] N M [0, 1] M R 2.1. f 0, f 1 : M N f0, f 1 : Hp (N) H p (M) 1. M, N H p (M) H p (M) π : M [0, 1] (x, t) x M, ι t : M x (x, t) M [0, 1] f i = F ι i ι 0 = ι 1 : Hp (M [0, 1]) H p (M) ι 0, ι 1 π π ι i = id M ι i π = id π ι i = id 2.1 2.2. t 0 [0, 1] K p : Ω p (M [0, 1]) Ω p 1 (M [0, 1]) ( 1) p 1 (dk p K p+1 d)ω = ω π ι t 0 ω Proof. ω Ω p (M [0, 1]) (x 1,..., x n, t) ω = I f I(x, t)dx I dt + J g J(x, t)dx J K p (ω) = I ( t t 0 f I (x, t)dt)dx I M t dk(ω) = f I (x, t)dt dx I + ( d x f I (x, t)dt) dx I I t 0 = t ( 1) p 1 f I (x, t)dx I dt + ( d x f I (x, t)dt) dx I I t 0 16

dω = I d x f I (x, t) dx I dt + J d x g J (x, t) dx J + g J dt dxj t = I d x f I (x, t) dx I dt + J d x g J (x, t) dx J + ( 1) p g J t dxj dt K p+1 dω = ( t t 0 d x f I (x, t)dt) dx I + ( 1) p (g J (x, t) g J (x, t 0 ))dx J (dk Kd)ω = ( 1) p 1 (ω J g J (x, t 0 )dx J ) = ( 1) p 1 (ω π ι t 0 ω) R n 1 ( ). H p R if p = 0 DR (Rn ) 0 otherwise Ω p cpt (M) Ω p cpt (M (0, 1)) Ωp 1 cpt (M) 2.2 K cpt : Ω p cpt (M (0, 1)) Ωp 1 cpt (M) ω = f I (x, t)dx I dt + g J (x, t)dx J Ω p cpt (M (0, 1)) K cpt ω = ( 1 0 f I (x, t)dt)dx I 2.2 g J dk cpt K cpt d = 0 K cpt π : H p cpt (M (0, 1)) Hp 1 cpt (M) 5. dk cpt K cpt d = 0 L : Ω p 1 cpt (M) Ωp cpt (M (0, 1)) φ C (0, 1) 1 0 φ(t)dt = 1 η Ω p 1 (M) cpt Lη = π η φ(t)dt π η dl = Ld L : H p 1 cpt (M) Hp cpt (M (0, 1)) K cpt L(η) = η L K cpt L K cpt (ω) = ( 1 0 f I (x, t)dt)dx I φ(t)dt (13) 17

2.3. π : H p cpt (M (0, 1)) Hp 1 cpt (M) L K cpt 2.2 H : Ω p cpt (M (0, 1)) Ωp 1 cpt (M (0, 1)) H(ω) = ( t 0 f I (x, t)dt t 0 φ(t)dt 1 0 f I (x, t)dt)dx I Hω 2.2 (13) 2.4. dh(ω) H(dω) = ( 1) p 1 (ω L K cpt (ω)) 6. 2.4 1 ( ). H p R if p = n cpt (Rn ) = 0 otherwise 2.2 M U, V M = U V j U : U V U, j V : U V V, i U : U M, i V : V M Ω p (M) i U i V Ω p (U) Ω p (V ) j U j V Ω p (U V ) (14) i U i V j U j V Im(i U i V ) ker(j U j V ) (ω U, ω V ) ker(ju j V ) j U j V (ω U ω V ) = ju (ω U) jv (ω V ) = 0 U, V ω U, ω V U V U ω U V ω V M ω i U i V (ω) = (ω U, ω V ) Im(i U i V ) = ker(j U j V ) (14) Ω p (U) Ω p (V ) 0 Ω p (M) i U i V Ω p (U) Ω p (V ) j U j V Ω p (U V ) 0 (15) Ω p (M), Ω p (U V ) Ω p (M) i U i V Ωp (U V ) ju j V 2.5. (15) Ω p (M), Ω p (U) Ω p (V ), Ω p (U V ) 18

1. (15) 3 Proof. i U i V j U j V U, V χ U, χ V χ U, χ V U, V ω Ω p (U V ) χ U ω V χ V ω U ω = (ju j V )(χ V ω, χ U ω ) (15) p Ω p Ω p+1 Ω p+2 d 2 = 0 Im ker......... d d 0 0 0 Ω p 1 (M) i U i V Ω p 1 (U) Ω p 1 (V ) ju j V d Ω p 1 (U V ) d d Ω p (M) i U i V Ω p (U) Ω p (V ) ju j V d Ω p (U V ) d d Ω p+1 (M) i U i V Ω p+1 (U) Ω p+1 (V ) ju j V d Ω p+1 (U V ) 0 0 0 d... d... d... 2.6 ( ). δ δ H p 1 DR (M) δ H p DR (M) δ H p+1 DR (M) i U i V H p 1 DR (U) Hp 1 DR (V ) i U i V H p DR (U) Hp 1 DR (V ) i U i V H p+1 DR (U) Hp 1 DR (V ) ju j V H p 1 DR (U V ) ju j V H p DR (U V ) j U j V H p+1 DR (U V ) Proof. H p DR (U) Hp 1 DR (V ) (15) (j U j V ) i U i V = 0 Im ker ([ω U], [ω V ]) ker(ju j V ) η Ωp 1 (U V ) U V ω U ω V = dη (15) η = η U η V ω U dη U, ω V dη V U V ω Ω p (M) i U i V ([ω]) = ([ω U dη U, ω V dη V ]) = ([ω U, ω V ]) δ [η] H p 1 DR (U V ) (15) ju j V (η U, η V ) = η U V dη U dη V = dη = 0 (15) dη U, dη V M p-form ω Ω p (M) U, V dω = 0 H p 1 DR (U V ) [η] [ω] Hp (M) δ η η U, η V δ δ (ju j V ) = 0 [η] ker δ 19

ω = dα η U α, η V α (ju j V )([η U α], [η V α]) = [η U η V ] = [η] [η] Im(jU j V ) 7. 2.6 H p DR (M) 2.6 U, V M 2.1 1 2.1. S n n B N n B S B S B N S n 1 S n 1 (0, 1) k : S n B S B N k : H p DR (B N B S ) H p DR (Sn 1 ) p 1 2.1 1 H p DR (B N) = H p DR (B S) = 0 H p DR (B N) H p DR (B S) = 0 H p+1 DR (Sn ) j U j V H p DR (B N B S ) = H p DR (Sn 1 ) i U i V H p+1 DR (B N) H p+1 DR (B S) = 0 δ H p DR (Sn 1 ) H p+1 DR (Sn ) 2 p n H p DR (Sn ) HDR 1 (Sn p+1 ) H 1 H 0 DR (Sn ) i U i V H 0 DR (B N) H 0 DR (B S) j U j V H 0 DR (B N B S ) δ H 1 DR (Sn ) i U i V H 1 DR (B N) H 1 DR (B S) = 0 0 H 0 (S n ) = R, n > 0, H 0 (S 0 ) = R 2 H 1 (S n ) = 0, n > 1, H 1 (S 1 ) = R n > 0 H p R if p = 0, n DR (Sn ) = 0 otherwise 2.2. n T n 1.3 dim H p DR (T n ) = nc p n = 1 2.1 T n = T n 1 S 1 S 1 I N, I S U = T n 1 I N, V = T n 1 I S T n T n 1 U, V H p DR (U) Hp DR (T n 1 ) H p DR (V ) U V = T n 1 (I N I S ) W 0, W 1 W i T n 1 f i : T n 1 W i U f 0, f 1 H p DR (U) Hp DR (T n 1 ) α (α, α) H p DR (T n 1 ) H p DR (T n 1 ) H p DR (W 0) H p DR (W 1) = H p DR (U V ) 20

V j p := ju j V Hp (T n 1 ) H p (T n 1 ) j p : H p (U) H p (U) (α, β) (α β, α β) H p DR (U V ) p = { (α, β) H p DR (T n 1 ) H p DR (T n 1 ); α = β } Im j p = ker j p = p H p DR (T n 1 ) 2.6 H p DR (T n ) Im i p ker i p = ker j p Im δ p 1 ker j p H p 1 (U V )/ Im j p 1 H p DR (T n 1 ) H p 1 DR (T n 1 ) HDR (T n 1 ) HDR (T n ) b n,p = dim H p (T n ) b n,p = b n 1,p + b n 1,p 1 b n,p = n C p 2.6 M ( )M {U α } U α1 U α2 U αk R n 2.7. M Proof. M U 1,..., U k k k = 1 2.1 1 U = U k, V = k 1 i=1 U i U, V U V = k 1 i=1 U k U i U V 2.6 H p 1 DR (U V ) Hp DR (M) Hp DR (U) Hp DR (V ) dim H p DR (M) dim Hp DR (U)+dim Hp DR (V )+dim Hp 1 DR (U V ) M x T x M g x g x x x g x g (M, g) 8. (M, g) γ : [a, b] M b L(γ) = g( dγ dt, dγ dt )dt a d(x, y) := inf {L(γ); γ : [0, 1] M, γ(0) = x, γ(1) = y} r B(x, r) d(x, y) x, y γ : [a, b] M t 0 [a, b] t 0 γ 21

2.2 ( ). C (M, g) x, y C x, y γ γ x, y C 2.3. R n S n x v T x M γ v (0) = x, γ v (0) = v γ v t = 0 γ t D r T x M r r exp x : D r v γ v (1) M D r 2.8. (M, g) x r > 0 (1) exp x : D r B(x, r) (2) B(x, r) M r > 0 x 1. M Proof. 2.8 x r > 0 ρ = r/2 B(x, ρ) y, z d(y, z) < r M U 1 = B(x 1, ρ),..., U k = B(x k, ρ) V = U i1 U il x V V B(x, r) W = exp 1 x (V ) exp x : W V V w W 0, w T x M L 2.8 x, exp x (w) L W T x M R n 9. U R n R n 2.3 HDR n (M) M n H n (M) [ω] ω R M M 22

2.9. M n n-formω ω = 0 ω = dη ω M 2.3 1 Hcpt(R n n ) = R ω Ω n cpt(r n ) R ω = 0 η Ω n 1 n cpt (Rn ) ω = dη M = S n H n (S n ) = R S n R n U N, U S χ N, χ S ω N = χ N ω, ω S = χ S ω U N, U S a := ω N = U N ω S U S (16) a = 0 x U N U S V U N U S V φ(x)dx 1... dx n = 1 V V bump function α = aφ(x)dx 1 dx n V ω N, ω S ω N α, ω S +α a = 0 ω N α = dη N, ω S + α = dη S U N, U S (n 1)-form η = η N + η S ω N, ω S (16) U N U S M R n U 1,..., U k χ i ω i = χ i ω a i = 0, a i = i U i ω i a i = 0 U 1,..., U k U i U j U i, U j Γ M Γ v 1,..., v k Γ a 1,..., a k v i, v j a i, a j α a i + α, a j α Γ 23

10. (1) Γ T (2) Proof. ( 2.9) Γ T T v i v i v j (a i, a j ) (0, a i + a j ) v i T a i 2.10. M n σ HDR n (M) = 0, Hn cpt(m) = R 2.9 n M, N f : M N HDR n (M) = Hn DR (N) = R [ω] 0 Hn DR (N) M deg f := f ω N ω deg f f f q N f 1 (q) x df x : T x M T q N df x f x ε x = 1 ε x = 1 f 1 (q) M f 1 (q) 2.11. M, N, f, q deg f = x f 1 (q) ε x Proof. f 1 (q) = {x 1,..., x l } x i U i q V i f f : U i V i V i V i W i = f 1 (V ) U i f W i, V f 1 (V ) = i W i V ω f ω = f ω = ε i ω M i W i i V 3 3.1 n M k Σ M Hcpt(M) k [ω] ω R 24 Σ

Σ (Hcpt(M)) k [Σ] Hcpt(M), k H n k DR (M) Hcpt(M) k H n k DR (M) ([ω], [η]) ω η P D : H n k DR (M) (Hk cpt(m)) [Σ] = P D 1 [Σ] H n k (M) DR 3.1 ( ). M P D : H n k DR (M) (Hk DR (M)) M {U α } N α=1 2.7 P D : H n k DR (M) (Hk cpt(m)) M R n 2.3 1 P D 2.9 k = n N M U 0 U 1 U 0 U 1 i U 1 U 0 3.2. U, V M 0 Ω p cpt (U V ) j Ω p cpt (U) Ωp cpt (V ) i Ω p cpt (U V ) 0 j = i U U V iv U V, i = i U V U + i U V V Proof. χ U, χ V ω Ω p cpt (U V ) (χ Uω, χ V ω) Ω p cpt (U) Ωp cpt (V ) (15) 2.6 3.3 ( ). δ H p 1 cpt (U V ) δ H p cpt (U V ) δ H p+1 cpt (U V ) j H p 1 cpt (U) Hp 1 cpt (V ) j H p cpt (U) Hp 1 cpt (V ) j H p+1 cpt (U) Hp 1 cpt (V ) i H p 1 cpt (U V ) i H p cpt (U V ) i H p+1 cpt (U V ) H p DR (U V ) P D i U i V H p DR (U) Hp DR (V ) ju j V H p DR (U V ) δ H p+1 DR (U V ) cpt P D (H n p cpt (U V )) i (H n p (U) H p 1 j (V )) cpt P D ±P D (H n p cpt (U V )) δ (H n p 1 cpt (U V )) (17) 3.3 P D δ = ±δ P D 25

11. A, B f : A B f : B A ker f = {b B ; b (x) = 0 if x Im f} Im f = {a A ; a (x) = 0 if x ker f} 3.4. [ω] H p DR (U V ), [η] Hn p 1 cpt (U V ) δω η = ( 1) p 1 ω δ η U V U V Proof. δ U V ω = ω U ω V ω U Ω p (U), ω V Ω p (V ) U V 0 = dω = dω U dω V δω = dω U = dω V U V δω η = χ U dω U η + χ V dω V η U V U V ω U χ V ω U V dχ U ω U η dχ V ω V η = dχ U ω η U V δ η (χ U η, χ V η) (dχ U η, dχ V η) U V dχ U η = dχ V η δ η = dχ U η U V 5-3.1 12. A 1,..., A 5, B 1,..., B 5 A 1 A 2 A 3 A 4 A 5 f 1 f 2 f 3? f 4 f 5 B 1 B 2 B 3 B 4 B 5 f 3 f 1, f 2, f 4, f 5 f 3 Proof. ( 3.1) (17) 5- U, V, U V U V {U α } N α=1 N N = 1 N = m N = m + 1 U = U 1 U m, V = U m+1 U V = (U 1 U m+1 ) (U m U m+1 ) m 5-26

3.2 n M n k Σ [Σ] (H n k [Σ] = P D 1 ([Σ] ) HDR k DR (M)) (M) Σ k- ω Σ ω Σ [η] H n k Σ η = M DR (M) η ω Σ (18) ω Σ k Σ [Σ] [Σ ] := ω Σ ω Σ M x Σ T x Σ T x M M g ν x Σ := T x Σ = {ξ T x M; g(ξ, v) = 0 v T x Σ} Σ ν x Σ = T x M/T x Σ p-form Σ 3.1 ( ). k E, B π : E B p B E p := π 1 (p) k p B p U πφ U (u, ξ) = u Φ U : U R k E U = π 1 (U) u U R n ξ Φ(u, ξ) π 1 (u) π : E B B k E (total space) B (base space) p M E p p 1. B E 3.1 ( ). E = B R k π : E B B B R k 3.2 ( ). n M T M = {(x, v); x M, v T x M} π : T M M n 27

3.3. Σ n k M n E = {(x, n); x Σ, n ν x Σ} π : E Σ Σ k νσ 13. p-form p T M π : E B U, V φ UV : U V GL n (R) Φ 1 V Φ U : (U V ) R k (u, ξ) (u, φ UV (u)ξ) (U V ) R k U α B {U α } α U α φ αβ : U α U β GL n (R) GL + n (R) = {A GL n (R); det A > 0} U α φ αβ GL + n (R) E B E E x R n 3.4 ( ). M n T M 1.9 n T M M Σ M νσ E B T M M g νσ M D r (νσ) = {n νσ; n g < r} D r (νσ) νσ 3.5 ( ). M Σ M Σ νσ T T Σ Proof. M g exp exp : νσ (x, n) exp p n M exp Σ νσ Σ M exp Σ r > 0 D r (νσ) T M 3.5. 3.5 1 dim Σ + 1 = dim M Σ, M νσ 1.9 Σ Σ R 28

Σ n k k-form ω Σ ω Σ T νσ Σ R k 2.3 π : H p+k cpt (Σ Rk ) H p DR (Σ) (19) π R n p = 0 π ([ω]) = [1] u Hcpt(Σ k R k ) u ω ω = 1 R k ω u Σ, R k x, y R k φ φ(y)dy R 1... dy k = 1 k ω(x, y) = φ(y)dy 1 dy k 3.6. Σ νσ u ω (18) Proof. (18) ω D r R k φ r ω r = φ r (y)dy 1 dy k η ω r = φ M Σ R r (y)η(x, y)dx dy k r η ω r η = φ M Σ Σ R r (y)(η(x, y) η(x, 0))dx dy 0 as r 0 k E B 3.6 2.3 (19) B E Ω p fiber (E) Hp fiber (E) π : E k B π U B E U U R k U x 1,..., x n k R k t 1,..., t k ω Ω p+k (E) fiber ω = I ω I 0(x, t)dx I dt 1 dt k + ω dx dt +... dx π : Ω p+k fiber (E) Ωp (B) π ω = ( ω0(x, I t)dt 1 dt k )dx I I 29

E V V R k (y, s) φ Φ x = φ(y), t = Φ(y)s dx i = j φ i dy j dt i = φ i j (x)dsj + dy +... ω ω = I ω I 0dx I det Φ(y)ds 1 ds k + ω dy ds +... det Φ > 0 π : H p+k fiber (U Rk ) H p (U) 2.3 π 0 Ω p+k fiber (E U V ) Ω p+k fiber (E U) Ω fiber (E V ) Ω p+k fiber (E U V ) 0 π π π 0 Ω p DR (U V ) Ωp DR (U) Ω DR(V ) Ω p DR (U V ) 0 H p+k fiber (E U V ) π H p+k fiber (E U) Ω fiber (E V ) π H p+k fiber (E U V ) π δ H p DR (U V ) Hp DR (U) Ω DR(V ) H p DR (U V ) δ 5-3.7. B k π : E B π H p+k cpt (E) Hp DR (B) p = 0 u Hcpt(E) k π u = [1] HDR 0 (B) E u ω 1 1. 3.6 νσ Proof. λ λ : νσ νσ ω u λ ω ω λ = λ ω ω λ = λ k ω 0 (x, λt)dt 1 dt k + λ k 1 (dx dt ) +... ω D 1 (νσ) ω λ D λ 1(νΣ) 3.6 (18) λ k λ ω Ω k cpt(e), η B Ω n k (B) π ω η Σ = ω π η B (20) B 30 M

B η = M ω π ι η ω η B = ι η ι : B E ω 14. (20) 2. E, B, u 3.7 U B ω U E U K U K ω U u ω Proof. V = M \ K M = U V E U, E V ω U, ω V U V U V U, V ω U ω V = dα α = α U α V α U, α V α U K ω U dα U, ω V dα V U V M = U V 3.3 3.2. Σ 1, Σ 2 M x Σ 1 Σ 2 T x M = T x Σ 1 + T x Σ 2 Σ 1, Σ 2 x Σ 1 Σ 2 Σ 1, Σ 2 dim M = n, dim Σ 1 = k, dim Σ 2 = n k x M U Σ 1 U = {(x 1,..., x k, 0,..., 0)}, Σ 2 U = {(0,..., 0, x k+1,..., x n )} (21) Σ 1 Σ 2 M, Σ 1, Σ 2 Σ 1, Σ 2 Σ 1 Σ 2 Σ 1, Σ 2 T 1, T 2 T 1 T 2 x Σ 1 Σ 2 Dr k D n k R k R n k M, Σ 1, Σ 2 (x 1,..., x n ) M (x 1,..., x k ) Σ 1 ε 1 (x) = ±1 (x k+1,..., x n ) Σ 2 ε 2 (x) = ±1 sgn(x) = ε 1 (x)ε 2 (x) 15. (21) sgn(x) 3.8. M Σ k 1, Σn k 2 [Σ 1 ] H n k (M), [Σ 2 ] H k (M) ω 1, ω 2 ω 1 ω 2 = sgn(x) M x Σ 1 Σ 2 r 31

Proof. ω 1, ω 2 νσ 1, νσ 2 x Σ 1 Σ 2 U x 2 U x φ α dx = 1 ω 1 = ε 1 (x)φ 1 (x k+1,..., x n )dx k+1 dx n, ω 2 = ( 1) k(n k) ε 2 (x)φ 2 (x 1,..., x k )dx 1 dx k ω 1 ω 2 = sgn(x) M x Σ 1 Σ 2 D k r D n k r φ 1 (z)φ 2 (y)dy dz = x Σ 1 Σ 2 sgn(x) 1. Σ n M (n 1) Σ T M [Σ] H 1 (M) 32