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❙②♠♣❧❡❝t✐❝ ♥✐❧♠❛♥✐❢♦❧❞s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✳

❱✐❝t♦r ▼✳ ❇✉❝❤st❛❜❡r

❜✉❝❤st❛❜❅♠✐✳r❛s✳r✉

❙t❡❦❧♦✈ ▼❛t❤❡♠❛t✐❝❛❧ ■♥st✐t✉t❡✱ ❘✉ss✐❛♥ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡s✱ ▼♦s❝♦✇ ❙t❛t❡ ❯♥✐✈❡rs✐t② ❖s❛❦❛ ❉❡❝❡♠❜❡r ✷✱ ✷✵✶✶✳

❆❜str❛❝t✳ ❚❤❡ t❛❧❦ ✐s ❞❡✈♦t❡❞ t♦ t❤❡ r❡♠❛r❦❛❜❧❡ t♦✇❡rs ♦❢ ❜✉♥❞❧❡s Mn → Mn−1 → · · · → S1, n 2, ✇✐t❤ ✜❜❡r t❤❡ ❝✐r❝❧❡ S1✳ ❚❤✐s t♦✇❡rs ❛r❡ ❞❡✜♥❡❞ ❜② t❤❡ ♥✐❧♣♦t❡♥t ❣r♦✉♣s ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ r❡❛❧ ❧✐♥❡✳ ❊❛❝❤ Mn✱ n 2✱ ✐s ❛ s♠♦♦t❤ ♥✐❧♠❛♥✐❢♦❧❞ ✇✐t❤ ❛ ✷✲❢♦r♠ ✇❤✐❝❤ ❣✐✈❡s ❛ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡ ♦♥ ❛♥② M2k✳ ❙✉❝❤ ♠❛♥✐❢♦❧❞s ♣❧❛② ❛♥ ✐♠♣♦rt❛♥t r♦❧❡ ✐♥ ❞✐✛❡r❡♥t ❛r❡❛s ♦❢ ♠❛t❤❡♠❛t✐❝s✳ ❲❡ ✇✐❧❧ ❞✐s❝✉ss t❤❡ ❞✐✛❡r❡♥t✐❛❧✲❣❡♦♠❡tr✐❝ ❛♥❞ ❛❧❣❡❜r♦✲t♦♣♦❧♦❣✐❝ r❡s✉❧ts ❛♥❞ ✉♥s♦❧✈❡❞ ♣r♦❜❧❡♠s✱ ❝♦♥❝❡r♥✐♥❣ t❤✐s ♠❛♥✐❢♦❧❞s✳

✶

• r♦✉♣s ♦❢ ♣♦❧②♥♦♠✐❛❧ tr❛♥s❢♦r♠❛t✐♦♥s✳

P✉t Ln = {px(t) = t + n

k=1 xktk+1, xk ∈ R}✳

❲❡ ❤❛✈❡ Ln ∼ = Rn : px(t) ⇒ x = (x1, . . . , xn)✳ ❲❡ ✇✐❧❧ ❝♦♥s✐❞❡r Ln ❛s t❤❡ n✲❞✐♠ ❣r♦✉♣ ♦❢ ♣♦❧②♥♦♠✐❛❧ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ r❡❛❧ ❧✐♥❡ R → R : t → px(t), ✇✐t❤ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥✿ x ∗ y = z✱ ✇❤❡r❡ (px ∗ py)(t) = pz(t) = py(px(t)) mod tn+2.

✷

❊①❛♠♣❧❡✳ ❋♦r n = 4✿ pz(t) = (px ∗ py)(t) = px(t) +

4

• k=1

ykpx(t)k+1 mod t6 : z1 = x1 + y1, z2 = x2 + 2x1y1 + y2, z3 = x3 + (2x2 + x2

1)y1 + 3x1y2 + y3

z4 = x4 + 2(x3 + x1x2)y1 + 3(x2 + x2

1)y2 + 4x1y3 + y4.

✸

◆✐❧♣♦t❡♥t ❣r♦✉♣ str✉❝t✉r❡ ♦♥ Rn✳ ❚❤❡ ❣r♦✉♣ Ln ∼ = Rn ❤❛s t❤❡ str✉❝t✉r❡ ♦❢ ♥✐❧♣♦t❡♥t ❣r♦✉♣ ✇✐t❤ t❤❡ ✉♣♣❡r ❝❡♥tr❛❧ s❡r✐❡s Ln

n ⊂ · · · ⊂ Ln q ⊂ · · · ⊂ Ln 0 = Ln,

✇❤❡r❡ Ln

n = {0 ∈ R}✱

Rn−q ∼ = Ln

q = {px(t) = t + n

• k=q+1

xktk+1}. ❲❡ ❤❛✈❡ Ln

q = {x ∈ Ln | ∀y ∈ Ln :

[x, y] ∈ Ln

q+1}

❛♥❞ Ln

q /Ln q−1 ∼

= R ✐s t❤❡ ❝❡♥t❡r ♦❢ Ln/Ln

q ✱ q = 0, . . . , n − 1✳

✹

❚❤❡ ❝❛♥♦♥✐❝❛❧ ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥✳ ❚❤❡ ❧❡❢t ♠✉❧t✐♣❧✐❝❛t✐♦♥ ∗ ❣✐✈❡s t❤❡ ❝❛♥♦♥✐❝❛❧ ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥ (x : v → x ∗ v) : ρ : Ln → GT(n + 1) : ρ(px(t))

• 1

v

• =
• 1

x ∗ v

• ✐♥t♦ t❤❡ ❣r♦✉♣ ♦❢ ❧♦✇❡r tr✐❛♥❣✉❧❛r (n + 1) × (n + 1)✲♠❛tr✐❝❡s ✇✐t❤

♦♥❡s ♦♥ t❤❡ ❞✐❛❣♦♥❛❧✿ ρ(px(t)) = X = (xik), i, k = 0, . . . , n, ✇❤❡r❡ xi,k = [px(t)k+1]i+1 ✐s t❤❡ ❝♦❡✣❝✐❡♥t ♦❢ ti+1 ✐♥ px(t)k✳

✺

❊①❛♠♣❧❡✳ ❋♦r n = 4✿ ρ(px(t))

• 1

v

• =

       

1 x1 1 x2 2x1 1 x3 2x2 + x2

1

3x1 1 x4 2(x3 + x1x2) 3(x2 + x2

1)

4x1 1

               

1 v1 v2 v3 v4

       

.

✻

❉❡❢♦r♠❛t✐♦♥ t♦ t❤❡ st❛♥❞❛r❞ ❣r♦✉♣ str✉❝t✉r❡✳ ❚❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ∗ ♦♥ Rn ❝❛♥ ❜❡ ✇r✐tt❡♥ ❞♦✇♥ ❛s x ∗ y = x + y + A(x)y, ✇❤❡r❡ A(x) = (aik(x)) ✐s t❤❡ ❧♦✇❡r tr✐❛♥❣✉❧❛r (n × n)✲♠❛tr✐① ✇✐t❤ ③❡r♦s ♦♥ t❤❡ ❞✐❛❣♦♥❛❧ ❛♥❞ aik(x) = xi,k = [px(t)k+1]i+1, i = k.

✼

❆♥② ❧✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ B : Rn → Rn ♦❢ ❝♦♦r❞✐♥❛t❡s ✐♥ Rn ❜② B ∈ GL(n, R) ❣✐✈❡s ❛ tr❛♥s❢♦r♠❡❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦♥ Rn✿ x ∗B y

def

= B−1((Bx) ∗ (By)) = = B−1(Bx + By + A(Bx)By) = = x + y + (B−1A(Bx)B)y. ■♥ t❤❡ ❝❛s❡ ♦❢ ❛ s❝❛❧❛r ♠❛tr✐① τE✱ ✇❡ ♦❜t❛✐♥ x ∗τ y = x + y + A(τx)y. ❚❤✐s ❣✐✈❡s ❛ ❞❡❢♦r♠❛t✐♦♥ ♦❢ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✯ ✭τ = 1✮ t♦ t❤❡ st❛♥❞❛r❞ ❛❞❞✐t✐♦♥ ✭τ = 0✮ ♦♥ Rn✳

✽

❊①❛♠♣❧❡✳ ❋♦r n = 4✿ x ∗ y = x + y + τA1(x)y + τ2A2(x)y, ✇❤❡r❡ A1(x) =

    

2x1 2x2 3x1 2x3 3x2 4x1

     ,

A2(x) =

    

x2

1

2x1x2 3x2

1

     .

✾

❈♦❝♦♠♣❛❝t ❧❛tt✐❝❡s✳ ❚❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ∗ ❣✐✈❡s t❤❡ ❢r❡❡ ❛❝t✐♦♥s ♦❢ Ln ♦♥ Rn✿ ❚❤❡ ❧❡❢t s❤✐❢t v → x ∗ v ❣✐✈❡s ❛ ❧✐♥❡❛r ❛❝t✐♦♥ ρ, ❚❤❡ r✐❣❤t s❤✐❢t v → v ∗ x ❣✐✈❡s ❛ ♥♦♥✲❧✐♥❡❛r ❛❝t✐♦♥. ▲❡t ✉s ❝♦♥s✐❞❡r t❤❡ ❝❛♥♦♥✐❝❛❧ ❧❛tt✐❝❡✿ Γn = {px(t) ∈ Ln : xi ∈ Z} ✇✐t❤ t❤❡ ✉♣♣❡r ❝❡♥tr❛❧ s❡r✐❡s✿ Γn

n ⊂ · · · ⊂ Γn q ⊂ · · · ⊂ Γn 0 = Γn.

❚❤✐s ❧❛tt✐❝❡ Γn ∼ = Zn ✐s ❝♦❝♦♠♣❛❝t ✭✉♥✐❢♦r♠✮✳

✶✵

◆✐❧♠❛♥✐❢♦❧❞s✳ ❲✐t❤ r❡s♣❡❝t t♦ t❤❡ r✐❣❤t s❤✐❢ts ✇❡ ♦❜t❛✐♥ ❛ s♠♦♦t❤ ❝❧♦s❡❞ ❛♥❞ ❝♦♠♣❛❝t ♥✐❧♠❛♥✐❢♦❧❞ Mn = Rn/Γn. ❚❤❡ t❛♥❣❡♥t ❜✉♥❞❧❡ ♦❢ Mn ✐s T(Mn) = Rn ×Γn Rn → Mn = Rn/Γn ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❧✐♥❡❛r ❛❝t✐♦♥ ρ ✭❧❡❢t s❤✐❢t✮ ♦♥ ❛ ✜❜❡r Rn✳

✶✶

❲❡ ❤❛✈❡ t❤❡ t♦✇❡rs ♦❢ ❣r♦✉♣s Ln → Ln−1 → · · · → L1, Γn → Γn−1 → · · · → Γ1 ❛♥❞ t❤❡ ✐♥❞✉❝❡❞ t♦✇❡r Mn → Mn−1 → · · · → M1 = S1 ♦❢ ❜✉♥❞❧❡s Mn → Mn−1 ✇✐t❤ t❤❡ ✜❜❡r S1✳ ❋♦r ❡❛❝❤ n t❤❡ ♠♦♥♦♠♦r♣❤✐s♠ ❤♦❧❞s in : L1 → Ln : in(x1) = (x1, . . . , xk

1, . . . , xn 1).

■ts ❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ t❤❡ ♣r♦❥❡❝t✐♦♥ Ln → L1 ✐s t❤❡ ✐❞❡♥t✐t② ♠❛♣✳ ❚❤✉s ❢♦r ❡❛❝❤ n t❤❡ ❜✉♥❞❧❡ M1 → S1 ✇✐t❤ t❤❡ ✜❜❡r Ln

1/Γn 1

❤❛s ❛ s❡❝t✐♦♥✳

✶✷

▲❡❢t ✐♥✈❛r✐❛♥t ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs✳ ▲❡t ✉s ✜① t❤❡ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ R[x1, . . . , xn] ❛s t❤❡ r✐♥❣ ♦❢ ❢✉♥❝t✐♦♥s ♦♥ Ln ∼ = Rn✳ P✉t ❢♦r f(x) ∈ R[x1, . . . , xn] Ry

xf(x) def

= f(x ∗ y) =

• |I|0

DI(f(x))yI ✇❤❡r❡ Ry

x ✐s t❤❡ r✐❣❤t s❤✐❢t ♦♣❡r❛t♦r✱

I = (i1, . . . , in) ❛♥❞ yI = yi1

1 . . . yin n ✳

❋r♦♠ t❤❡ ❛ss♦❝✐❛t✐✈✐t② ❡q✉❛t✐♦♥ Ry

xRz x = Rz yRy x ✇❡ ❤❛✈❡

• |I|0
• |J|0

DIDJf(x)yJzI =

• |K|0

DKf(x)(y ∗ z)K.

✶✸

❊①❛♠♣❧❡ n = 3✳ ❲❡ ❤❛✈❡ D0f(x) = f(x)✱ D(1,0,0) = ∂ ∂x1 + 2x1 ∂ ∂x2 + (2x2 + x2

1) ∂

∂x3 , D(0,1,0) = ∂ ∂x2 + 3x1 ∂ ∂x3 , D(0,0,1) = ∂ ∂x3 . D(1,0,0)D(0,1,0) = D(1,1,0) + 2D(0,0,1), D(0,1,0)D(1,0,0) = D(1,1,0) + 3D(0,0,1).

✶✹

❚❤❡ ❛❧❣❡❜r❛ An ❣❡♥❡r❛t❡❞ ❜② t❤❡ ♦♣❡r❛t♦rs DI ✐s t❤❡ ❛❧❣❡❜r❛ ♦❢ ❛❧❧ ❧❡❢t ✐♥✈❛r✐❛♥t ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs ♦♥ R[x1, . . . , xn] ❢♦r t❤❡ ❧❡❢t s❤✐❢t Lz

x✿

Lz

xf(x) = f(z ∗ x),

t❤❛t ✐s Lz

xDIf(x) = DILz xf(x)

❢♦r z ❛s ♣❛r❛♠❡t❡r✳

✶✺

❆❧❣❡❜r❛ ♦❢ t❤❡ ❧❡❢t ✐♥✈❛r✐❛♥t ♦♣❡r❛t♦rs✳ ❚❤❡ ❛❧❣❡❜r❛ An ✐s ♠✉❧t✐♣❧✐❝❛t✐✈❡❧② ❣❡♥❡r❛t❡❞ ❜② t❤❡ ♦♣❡r❛t♦rs ξi = ∂i +

• xi,q∂q, i = 1, . . . , n,

✇❤❡r❡ ∂i =

∂ ∂xi✱ ❛♥❞ xiq ✐s t❤❡ ❝♦❡✣❝✐❡♥t ♦❢ tq+1 ✐♥ t❤❡ ♣♦❧②♥♦♠✐❛❧

px(t)i+1✱ ❛s ❜❡❢♦r❡✳ ❚❤❡ ❝♦♠♠✉t❛t♦rs ♦❢ t❤✐s ♦♣❡r❛t♦rs ❛r❡ [ξi, ξj] = (j − i)ξi+j ✇✐t❤ ξq = 0 ✐❢ q > n✳ ❊①❛♠♣❧❡✳ ❋♦r n = 3 A3 = R[ξ1, ξ2, ξ3]/([ξ1, ξ2] = ξ3, [ξ1, ξ3] = [ξ2, ξ3] = 0).

✶✻

❚❤❡ ♦♣❡r❛t♦rs {ξi} ❝♦♥st✐t✉t❡ ❛ ❜❛s✐s ✐♥ t❤❡ ▲✐❡ ❛❧❣❡❜r❛ Ln ♦❢ t❤❡ ❧❡❢t ✐♥✈❛r✐❛♥t ✈❡❝t♦r ✜❡❧❞s ♦♥ t❤❡ ❣r♦✉♣ Ln✱ ❛♥❞ t❤❡ ♦♣❡r❛t♦r ξm ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ♦♥❡✲♣❛r❛♠❡t❡r s✉❜❣r♦✉♣ φm(s) ♦❢ ♣♦❧②♥♦♠✐❛❧s

• ϕm(t; s) = t(1 − mstm)− 1

m

mod tn+2

• ,

m = 1, 2, . . . , n. ❲❡ ❤❛✈❡ ϕm(t; s) = t+stm+1+

• k2

(1+m)(1+2m) . . . (1+(k−1)m)sk k!tkm+1. ◆♦t❡ φm(t, 1) / ∈ Γn ❢♦r m > 1✱ ❜✉t ϕm(t; m) = ϕm(t; 1)m ∈ Γn✳

✶✼

❊①❛♠♣❧❡✳ ❋♦r n = 4 ϕ1(t; s) = t + st2 + s2t3 + s3t4 + s4t5, ϕ2(t; s) = t + st3 + 3 2s2t5, ϕ3(t; s) = t + st4, ϕ4(t; s) = t + st5. ϕ1(t; 1) = e1 ∗ e2 ∗ e−2

3

∗ e6

4,

✇❤❡r❡ e−1

3 (e3(t)) = t✳

✶✽

❈♦❤♦♠♦❧♦❣② r✐♥❣ ♦❢ ❛ ❞✐✛❡r❡♥t✐❛❧ ❣r❛❞❡❞ ❛❧❣❡❜r❛✳ ❆ ❞✐✛❡r❡♥t✐❛❧ ❣r❛❞❡❞ ❛❧❣❡❜r❛ ✭❞✳ ❣✳ ❛✳✮ (C, d) ✐s ❛ ❣r❛❞❡❞ ❛❧❣❡❜r❛ C =

• p0

Cp ✇✐t❤ ❛ ❞✐✛❡r❡♥t✐❛❧ d : C → C ♦❢ ❞❡❣r❡❡ ✶✱ ✐✳ ❡✳ d(Cp) ⊂ Cp+1 ❛♥❞ d2 = 0✱ s✉❝❤ t❤❛t a · b = (−1)pqba ❢♦r a ∈ Cp✱ b ∈ Cq✱ dab = (da)b + (−1)pa(db) ❢♦r a ∈ Cp✳ P✉t ZpC = ker(d : Cp → Cp+1) − ❝♦❝②❝❧❡s ❣r♦✉♣, BpC = Im(d : Cp−1 → Cp) − ❝♦❜♦✉♥❞❛r✐❡s ❣r♦✉♣, ❛♥❞ HpC = ZpC/BpC − ❝♦❤♦♠♦❧♦❣② ❣r♦✉♣. ❚❤❡♥ H∗C =

• p0

HpC ✐s ❛ ❞✳❣✳❛✳ ✭✇✐t❤ d = 0✮ ✖ ❝♦❤♦♠♦❧♦❣② r✐♥❣ ♦❢ C✳

✶✾

❊①❛♠♣❧❡✳ ▲❡t X ❜❡ ❛ s♠♦♦t❤ n✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♠♣❛❝t ♠❛♥✐❢♦❧❞✳ ❚❤❡♥ ✇❡ ❤❛✈❡ ❛ ❞✳❣✳❛✳ ♦❢ s♠♦♦t❤ r❡❛❧ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠s C(X) =

• p0

Cp(X). ■♥ ❛ ❝♦♦r❞✐♥❛t❡ ♥❡✐❣❤❜♦✉r❤♦♦❞ U ⊂ X ✇❡ ❤❛✈❡ ❢♦r ω ∈ Cp(X) ω =

• ui1...ip(x)dxi1 ∧ · · · ∧ dxip,

x = (xi1, . . . , xi1) ∈ U, dω =

• i1<i2<···<ip

dui1...ip ∧ dxi1 ∧ · · · ∧ dxip = =

• i1<i2<···<ip

∂ ∂xi0ui1...ipdxi0 ∧ dxi1 ∧ · · · ∧ dxip ❛♥❞ H∗C(X) = H∗(X; R)✳

✷✵

❉✐✛❡r❡♥t✐❛❧ ❣r❛❞❡❞ ❛❧❣❡❜r❛ ♦❢ t❤❡ ❧❡❢t ✐♥✈❛r✐❛♥t ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠s ♦♥ t❤❡ ♥✐❧♠❛♥✐❢♦❧❞✳ ▲❡t ω1, . . . , ωn ❜❡ t❤❡ ❜❛s✐s ♦❢ t❤❡ ❧❡❢t ✐♥✈❛r✐❛♥t ❞✐✛❡r❡♥t✐❛❧ ✶✲❢♦r♠s ♦♥ Ln ❞✉❛❧ t♦ t❤❡ ❜❛s✐s {ξi}✳ ▲❡t ω =

n

• i=1

ωiξi ❜❡ t❤❡ ▼❛✉r❡r✲❈❛rt❛♥ ❢♦r♠ t❛❦✐♥❣ ✈❛❧✉❡s ✐♥ t❤❡ ▲✐❡ ❛❧❣❡❜r❛ Ln ♦❢ ✈❡❝t♦r ✜❡❧❞s ξi✱ i = 1, . . . , n✳

✷✶

❚❤❡ ▼❛✉r❡r✕❈❛rt❛♥ ❡q✉❛t✐♦♥ dω = −1 2[ω, ω] ✐♥ ♦✉r ❝❛s❡ t❛❦❡s t❤❡ ❢♦r♠ dωq =

• {(k,l): k>l>0, k+l=q}

(k − l) ωk ∧ ωl. ✭✶✮ ❍❡r❡ [ω, ω](ζ1, ζ2) = [ω(ζ1), ω(ζ2)]✳ ◆♦t❡ t❤❛t dω1 = dω2 = 0 ❛♥❞ ✭✶✮ ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ n✳ ❊①❛♠♣❧❡s✿ dω3 = ω2∧ω1✱ dω4 = 2ω3∧ω1✱ dω5 = 3ω4∧ω1+ω3∧ω2✳

✷✷

❇✐❣r❛❞❡❞ ❝♦❤♦♠♦❧♦❣② r✐♥❣✳ ❲❡ ❤❛✈❡ H∗(Mn; R) = H(Λ(ω1, . . . , ωn), d) ✇❤❡r❡ Λ( ) ✐s t❤❡ ❡①t❡r✐♦r ❛❧❣❡❜r❛✱ ❛♥❞ d ❤❛s t❤❡ ❢♦r♠ ✭✶✮✳ ❙❡t bideg ωq = (1, −2q)✳ ■t ❢♦❧❧♦✇s ❢r♦♠ ✭✶✮ t❤❛t t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❝♦♠♣❧❡① (Λ(ω1, . . . , ωn), d) ❝❛♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ❛s ❛ s✉♠ ♦❢ ❞✐✛❡r❡♥t✐❛❧ s✉❜❝♦♠♣❧❡①❡s Λ0 +

n

• q=1

(Λ−2q, d), ✇❤❡r❡ Λ0 = R ❛♥❞ (Λ−2q, d) ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ ❢♦r♠s ωi1 ∧ · · · ∧ ωis, s = 1, . . . , n, i1 > i2 > · · · > is > 0, i1 + · · · + is = q.

✷✸

❋♦r ❛♥② n 2 ✇❡ ❤❛✈❡✿ H1(Mn; R) = H1,−2(Mn; R) + H1,−4(Mn; R) = R ⊕ R ✇✐t❤ t❤❡ ❣❡♥❡r❛t♦rs [ω1] ❛♥❞ [ω2] ❝♦rr❡s♣♦♥❞✐♥❣❧②✳ ❚❤✉s H1(Mn; Z) = Z + Z✱ n 2✳ ❚❤❡ r✐♥❣ H∗(Mn, R) ❤❛s t❤❡ str✉❝t✉r❡ ♦❢ ❛ ❜✐❣r❛❞❡❞ r✐♥❣ R +

n

• s=1

s(2n+1−s)

• 2q=s(s+1)

Hs,−2q(Mn; R). ❙❡t n(s, q) = q − 1

2(s − 1)(s − 2)✳ ❋♦r ❛♥② k n(s, q) ✇❡ ❤❛✈❡

Hs,−2q(Mk; R) = Hs,−2q(Mk+1; R).

✷✹

❊①❛♠♣❧❡ ❢♦r n = 4✳ ❲❡ ❤❛✈❡✿ H∗(M4; R) = H∗(Λ(ω1, ω2, ω3, ω4), d), ✇❤❡r❡ dω1 = dω2 = 0✱ dω3 = ω2 ∧ ω1✱ dω4 = 2ω3 ∧ ω1✳ H∗(M4; R) = R +

4

• s=1

s(9−s)

• 2q=s(s+1)

Hs,−2q(M4; R).

✷✺

✶✵ ❞✐✛❡r❡♥t✐❛❧ s✉❜❝♦♠♣❧❡①❡s✳ q\s 1 2 3 4 1 1 ω1 2 ω2 3 ω3

1

→ ω2 ∧ ω1 4 ω4

2

→ ω3 ∧ ω1 5 ω4 ∧ ω1 ω3 ∧ ω2 6 ω4 ∧ ω2

−2

→ ω3 ∧ ω2 ∧ ω1 7 ω4 ∧ ω3

−1

→ ω4 ∧ ω2 ∧ ω1 8 ω4 ∧ ω3 ∧ ω1 9 ω4 ∧ ω3 ∧ ω2 10 ω4 ∧ ω3 ∧ ω2 ∧ ω1

✷✻

• ❡♥❡r❛t♦rs ♦❢ t❤❡ P♦✐♥❝❛r❡ ❞✉❛❧✐t②✳

dim 1 2 3 4 [ω1] ← → [ω4 ∧ ω3 ∧ ω2] [ω2] ← → [ω4 ∧ ω3 ∧ ω1] [ω1 ∧ ω4]

• [ω2 ∧ ω3]

1 ← → [ω4 ∧ ω3 ∧ ω2 ∧ ω1]

✷✼

❚♦r✐❝ ❜✉♥❞❧❡s✳ ❋♦r ❡❛❝❤ n ❛♥❞ q <

n+1

2

• t❤❡r❡ ❛r❡ ❡①❛❝t s❡q✉❡♥❝❡s

0 → Rq+1 → Ln+1 → Ln−q → 0, 0 → Zq+1 → Γn+1 → Γn−q → 0, ✇❤✐❝❤ ❣✐✈❡ ❛ s♠♦♦t❤ ❜✉♥❞❧❡ πq

n : Mn+1 → Mn−q

✇✐t❤ ✜❜r❡ t♦r✉s Tq+1✳

✷✽

❙②♠♣❧❡❝t✐❝ ♥✐❧♠❛♥✐❢♦❧❞s Mn✳ ❆ s♠♦♦t❤ ♠❛♥✐❢♦❧❞ M ✐s ❝❛❧❧❡❞ s②♠♣❧❡❝t✐❝ ✐❢ ✐t ❝❛rr✐❡s ❛ ♥♦♥❞❡❣❡♥❡r❛t❡ ❝❧♦s❡❞ 2✲❢♦r♠ Ω ✇❤✐❝❤ ✐s ❝❛❧❧❡❞ ❛ s②♠♣❧❡❝t✐❝ ❢♦r♠✳ ❈♦♥s✐❞❡r t❤❡ s♠♦♦t❤ ❜✉♥❞❧❡ ✇✐t❤ ✜❜r❡ ❝✐r❝❧❡ S1 πn = π0

n : Mn+1 → Mn.

❚❤❡ ❧❡❢t ✐♥✈❛r✐❛♥t ✶✲❢♦r♠ ωn+1 ✐s ❛ ❝♦♥♥❡❝t✐♦♥ ✐♥ t❤❡ ❜✉♥❞❧❡ πn✳ ❚❤❡ ❝✉r✈❛t✉r❡ ❢♦r♠ ♦❢ t❤✐s ❜✉♥❞❧❡ ✐s Ωn =

• {(k,l): k+l=n+1, k>l>0}

(k − l)ωk ∧ ωl ❛♥❞ ✇❡ ❤❛✈❡ π∗Ωn = dωn+1. ❚❤❡ ♥✐❧♠❛♥✐❢♦❧❞ M2n ✇✐t❤ t❤❡ ❢♦r♠ Ω2n ✐s s②♠♣❧❡❝t✐❝✳ ❈♦♥❥❡❝t✉r❡✳ Ωn ✐s ❛♥ ✐♥t❡❣❡r ❢♦r♠ ❢♦r ❛♥② n✳

✷✾

❊①❛♠♣❧❡✳ ❋♦r n = 3✱ q = 1 ✇❡ ❤❛✈❡ t❤❡ s♠♦♦t❤ ❜✉♥❞❧❡ π1

3 : M4 → M2 = T2

✇✐t❤ t❤❡ ✜❜r❡ T2 ❛♥❞ t❤❡ s②♠♣❧❡❝t✐❝ ❢♦r♠✿ Ω4 = 3ω4 ∧ ω1 + ω3 ∧ ω2. ❚❤❡ ❜❛s❡ ✐s t❤❡ s②♠♣❧❡❝t✐❝ ♠❛♥✐❢♦❧❞ ✇✐t❤ t❤❡ ❢♦r♠ Ω2 = ω2 ∧ ω1 ❛♥❞ (π1

3)∗Ω2 = 0✳

❚❤❡ ♠❛♥✐❢♦❧❞ M3 × S1 ✐s s②♠♣❧❡❝t✐❝ ✇✐t❤ t❤❡ ❢♦r♠ 2ω3 ∧ ω1 + ω2 ∧ dt.

✸✵

❚❤❡ ♠❛♥✐❢♦❧❞ M2n−1 ❤❛s t❤❡ ❢♦r♠ Ω2n−1 =

• {(k,l): k+l=2n, k>l>0}

(k − l)ωk ∧ ωl. ❋♦r n > 2 t❤❡ ❢♦r♠ ωn ✐s ♥♦t ❝❧♦s❡❞✱ t❤✉s t❤❡ ✷✲❢♦r♠ Ω = Ω2n−1 + ωn ∧ dt ✐s ♥♦t ❝❧♦s❡❞ ♦♥ M2n−1 × S1 ❜✉t Ωn ✐s ❝❧♦s❡❞ ❛♥❞ ❣✐✈❡s t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❝♦❝②❝❧❡ ♦♥ t❤✐s ♠❛♥✐❢♦❧❞✳

✸✶

◆♦♥❢♦r♠❛❧✐t② ♦❢ ♥✐❧♠❛♥✐❢♦❧❞s✳ ❆ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡① X ✐s ❝❛❧❧❡❞ ❢♦r♠❛❧ ✐❢ ✐ts r❛t✐♦♥❛❧ ❤♦♠♦t♦♣② t②♣❡ ✐s ❛ ❢♦r♠❛❧ ❝♦♥s❡q✉❡♥❝❡ ♦❢ ✐ts ❝♦❤♦♠♦❧♦❣② r✐♥❣✳ ❚❤❡♦r❡♠✳ ✭❋✳ ❊✳ ❆✳ ❏♦❤♥s♦♥✱ ❊✳ ●✳ ❘❡❡s✱ ✶✾✽✾✮ ■❢ G ✐s ❛ ♥✐❧♣♦t❡♥t ▲✐❡ ❣r♦✉♣ ❛♥❞ Γ ⊂ G ✐s ❛ ❞✐s❝r❡t❡ ❝♦❝♦♠♣❛❝t s✉❜❣r♦✉♣✱ t❤❡♥ G/Γ ✐s ❢♦r♠❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ G ✐s ❛❜❡❧✐❛♥✳ ❈♦r♦❧❧❛r②✳ ❚❤❡ s②♠♣❧❡❝t✐❝ ♥✐❧♠❛♥✐❢♦❧❞s M2m ❛r❡ ♥♦♥❢♦r♠❛❧✱ m 2✱ ❛♥❞ M2 = T 2 ✐s ❢♦r♠❛❧✳

✸✷

❘❡❛❧✐③✐♥❣ ♥✐❧♠❛♥✐❢♦❧❞s ❛s s②♠♣❧❡❝t✐❝ s✉❜♠❛♥✐❢♦❧❞s ♦❢ ❝♦♠♣❧❡① ♣r♦❥❡❝t✐✈❡ s♣❛❝❡s CP N✱ ❞❡♥♦t❡ ❜② Xm(N) t❤❡ s②♠♣❧❡❝t✐❝ ❜❧♦✇ ✉♣ ♦❢ CP N ❛❧♦♥❣ M2n✳ ❚❤❡♦r❡♠✳ ✭■✳❑✳ ❇❛❜❡♥❦♦✱ ■✳❆✳ ❚❛✐♠❛♥♦✈✱ ✶✾✾✾✮ ❋♦r m 2 ❛♥❞ N 2m + 1 t❤❡ s②♠♣❧❡❝t✐❝ ♠❛♥✐❢♦❧❞s Xm(N) ❛r❡ s✐♠♣❧② ❝♦♥♥❡❝t❡❞ ❛♥❞ ♥♦♥❢♦r♠❛❧✳ ❚❤❡ ♣r♦♦❢ ♦❢ t❤✐s r❡s✉❧t ♠❛❦❡s ✉s❡ ♦❢ t❤❡ ❢❛❝t t❤❛t ✐♥ t❤❡ ❝♦❤♦♠♦❧♦❣② r✐♥❣ H∗(M2n) t❤❡r❡ ❛r❡ ♥♦♥tr✐✈✐❛❧ ▼❛ss❡② ♣r♦❞✉❝ts✳

✸✸

❯♥✐✈❡rs❛❧ ♣r♦♣❡rt✐❡s ♦❢ Mn✳ ❚❤❡ ♠❛♥✐❢♦❧❞ Mn = K(Γn, 1) ✐s t❤❡ ❊✐❧❡♥❜❡r❣✲▼❛❝▲❛♥❡ s♣❛❝❡ ❛♥❞ t❤✉s ❢♦r ❛♥② CW✲❝♦♠♣❧❡① X [X, Mn] = H1(X, Γn). ❚❤❡ ♠❛♥✐❢♦❧❞ Mn ✐s t❤❡ ❝❧❛ss✐❢②✐♥❣ s♣❛❝❡ ❢♦r t❤❡ ❞✐s❝r❡t❡ ❣r♦✉♣ Γn✱ t❤❛t ✐s Mn = BΓn ❛♥❞ t❤✉s [X, Mn] ✐s t❤❡ s❡t ♦❢ ✐s♦♠♦r♣❤✐s♠ ❝❧❛ss❡s ♦❢ ♣r✐♥❝✐♣❛❧ Γn✲❜✉♥❞❧❡s ♦✈❡r ❛ CW✲❝♦♠♣❧❡① X❀ ✇❡ ❤❛✈❡ [X, Mn] = Hom(π1(X), Γn), Hk(Mn; Z) = Hk(Γn; Z), Hk(Mn; Z) = Hk(Γn; Z).

✸✹

❈❡❧❧✉❧❛r s✉❜❞✐✈✐s✐♦♥ ♦❢ Mn✳ ❈♦♥s✐❞❡r t❤❡ ❝❡❧❧✉❧❛r s✉❜❞✐✈✐s✐♦♥ (pt) = Mn

0 ⊂ Mn 1 ⊂ · · · ⊂ Mn n−1 ⊂ Mn n = Mn,

✇❤❡r❡ Mn

1 = ∨n i=1S1 i ✱ Mn k+1/Mn k = ∨Sk+1✱ Mn n/Mn n−1 = ∨Sn✳

❋♦r t❤❡ Z✲❤♦♠♦❧♦❣② ❣r♦✉♣s ♦❢ ♣❛✐r ✇❡ ♦❜t❛✐♥ t❤❡ ❡①❛❝t s❡q✉❡♥❝❡ 0 → H2(Mn) → H2(Mn/Mn

1) → ⊕n i=1Z → H1(Mn) → 0.

❯s✐♥❣ t❤❛t Mn = K(Γn; 1) ❛♥❞ Mn

1 = K(∨n i=1Z; 1)

❢♦r t❤❡ ❤♦♠♦t♦♣② ❣r♦✉♣s ♦❢ ♣❛✐r ✇❡ ♦❜t❛✐♥ t❤❡ ❡①❛❝t s❡q✉❡♥❝❡ 0 → Rn → ∨n

i=1Z → Γn → 0.

❍❡r❡ ∨n

i=1Z ✐s t❤❡ ❢r❡❡ ♣r♦❞✉❝t ♦❢ Z

❛♥❞ Rn = π2(Mn, Mn

1) ✐s ✐ts s✉❜❣r♦✉♣✳ ■t ✐s ❛ ❢r❡❡ ❣r♦✉♣✳

✸✺

❚❤❡ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❣❡♥❡r❛t♦rs ♦❢ t❤❡ ❣r♦✉♣ Γn ⊂ Ln ❛r❡ ek(t) = t + tk+1✱ k = 1, . . . , n✳ P✉t e0(t) = t✳ ◆♦t❡ ϕq(t; 1) = eq(t) ❢♦r q > [n

2]✳

■t ✐s ❝❧❡❛r t❤❛t ✐❢ eI = ei1

1 ∗· · ·∗ein n = e0 ✇❤❡r❡ I = (i1, . . . , in) ∈ Zn✱

t❤❡♥ I = 0✳ ❲❡ ❤❛✈❡ [ek, ek+2] = e2

2k+2 ∗ ei ∗ . . . ,

i > 2k + 2, k 1, [ek, ek+1] = e2k+1 ∗ ej ∗ . . . , j > 2k + 1, k 1. ❚❤✉s t❤❡ ❣r♦✉♣ H1(Mn; Z) = Γn/[Γn, Γn] ❤❛s ♦♥❧② ✷✲t♦rs✐♦♥✳

✸✻

❍♦♣❢✬s ✐♥t❡❣r❛❧ ❤♦♠♦❧♦❣② ❢♦r♠✉❧❛✳ ▲❡t G = F/R ❛♥❞ F ✐s ❛ ❢r❡❡ ❣r♦✉♣✳ ❚❤❡♥ H2(G, Z) ∼ = (R ∩ [F, F])/[F, R]. ❚❤✉s H2(Mn, Z) ∼ = (Rn ∩ [F n, F n])/[F n, Rn], ✇❤❡r❡ F n = ∨n

i=1Z ❛♥❞

0 → Rn → ∨n

i=1Z → Γn → 0,

❛♥❞ t❤❡r❡❢♦r❡ t♦ ❡❛❝❤ ❡❧❡♠❡♥t a ∈ H2(Mn, Z) ❝♦rr❡s♣♦♥❞s ❛♥ ❡❧❡♠❡♥t g = [a1, b1] · ... · [ag, bg] ∈ (Rn ∩ [F n, F n]).

✸✼

❊①❛♠♣❧❡ n = 3✳ Γ3 ❤❛s t❤❡ ❣❡♥❡r❛t♦rs e1✱ e2✱ e3 ❛♥❞ t❤❡ r❡❧❛t✐♦♥s [e1, e2] = e3, [e1, e3] = e0, [e2, e3] = e0. ❚❤✉s H1(M3, Z) = Γ3/[Γ3, Γ3] = Z ⊕ Z✳ ■♥ t❤✐s ❝❛s❡ F 3 ❤❛s t❤❡ ❣❡♥❡r❛t♦rs c1✱ c2✱ c3✱ R3 ❤❛s t❤❡ ❣❡♥❡r❛t♦rs r1✱ r2✱ r3 ❛♥❞ R3 → F 3 : r1 → [c1, c3], r2 → [c2, c3], r3 → [c1, c2]c−1

3 .

❲❡ ❤❛✈❡ r3 / ∈ [F 3, F 3]✳ ❚❤❡ ❣❡♥❡r❛t♦rs ♦❢ H2(M3; Z) = Z ⊕ Z ❝♦rr❡s♣♦♥❞ t♦ r1 ❛♥❞ r2✳

✸✽

❊①❛♠♣❧❡ n = 4✳ Γ4 ❤❛s t❤❡ ❣❡♥❡r❛t♦rs e1✱ e2✱ e3✱ e4 ❛♥❞ t❤❡ r❡❧❛t✐♦♥s [e1, e2] = e3 ∗ e4, [e1, e3] = e2

4,

[e1, e4] = e0, [ei, ej] = e0, i, j = 2, 3, 4. ❚❤✉s H1(M4, Z) = Γ4/[Γ4, Γ4] = Z ⊕ Z ⊕ Z2✳

✸✾

❈♦♥s✐❞❡r ❛♥ ♦r✐❡♥t❡❞ ✷✲❞✐♠❡♥t✐♦♥❛❧ s✉r❢❛❝❡ S2

g ♦❢ ❣❡♥✉s g✳

❲❡ ❤❛✈❡ S2

g = K(Gg, 1)✱

✇❤❡r❡ Gg = π1(S2

g ) ✐s t❤❡ ❣r♦✉♣ ✇✐t❤ t❤❡ ❣❡♥❡r❛t♦rs a1, b1, . . . , ag, bg

❛♥❞ ❛ s✐♥❣❧❡ r❡❧❛t✐♦♥ [a1, b1] · ... · [ag, bg] = 1, t❤❛t ✐s 0 → Z → ∨2g

i=1Z → Gg → 0.

❲❡ ❤❛✈❡ [S2

g , Mn] = Hom[Gg, Γn].

✹✵

❈♦r♦❧❧❛r②✳ ❊❛❝❤ ❡❧❡♠❡♥t a ∈ H2(Mn, Z)✱ n 2✱ ✐s r❡❛❧✐s❡❞ ❜② ❛ s♠♦♦t❤ ♠❛♣♣✐♥❣ fa : S2

g → Mn,

(fa)∗([S2

g ]) = a

❢♦r s♦♠❡ g✳ ❚❤❡ ❢♦r♠ Ωn ✐s ✐♥t❡❣❡r ✐❢ ❛♥❞ ♦♥❧② ✐❢ f∗

aΩn, [S2 g ] ∈ Z.

✹✶

▲❡t πn : Mn+1 → Mn ❜❡ ❛ s♠♦♦t❤ ❜✉♥❞❧❡ ✇✐t❤ t❤❡ ✜❜r❡ S1✳ ❉❡♥♦t❡ ❜② ξn+1 = ξn+1(πn) t❤❡ ✜❡❧❞ ♦❢ ✈❡❝t♦rs t❛♥❣❡♥t t♦ t❤❡ ✜❜❡r ♦❢ t❤✐s ❜✉♥❞❧❡✳ Pr♦❜❧❡♠✳ ❈❧❛ss✐❢② t❤❡ s❡q✉❡♥❝❡s ♦❢ s♠♦♦t❤ ♠❛♥✐❢♦❧❞s πn : Mn+1 → Mn, n 0, ✇✐t❤ t❤❡ ✜❜❡r S1✱ s✉❝❤ t❤❛t ✲ ❢♦r ❡❛❝❤ n > 1 t❤❡r❡ ❡①✐sts ❛♥ ✐♥t❡❣❡r ❝❧♦s❡❞ 2✲❢♦r♠ Ωn ♦♥ Mn s❛t✐s❢②✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥ π∗

nΩn = dωn+1,

✇❤❡r❡ ωn+1, ξn+1 = ||ξn+1||, ✲ ❢♦r ❡❛❝❤ ❡✈❡♥ n t❤❡ ❢♦r♠ Ωn ✐s ♥♦♥❞❡❣❡♥❡r❛t❡✳

✹✷

❚❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠ ✐s ❝❧♦s❡❧② r❡❧❛t❡❞ t♦ t❤❡ ♣r❡✈✐♦✉s ♦♥❡ ❛♥❞ ❤❛s s❡❧❢✲❝♦♥t❛✐♥❡❞ ✐♥t❡r❡st✿ Pr♦❜❧❡♠✳ ❋♦r t❤❡ t♦✇❡rs Mn → Mn−1 → · · · → S1 ♦❢ ✜❜r❛t✐♦♥s ❞❡s❝r✐❜❡❞ ❛❜♦✈❡ ❝❛❧❝✉❧❛t❡ t❤❡ ❝♦❤♦♠♦❧♦❣② r✐♥❣s H∗(Mn; k) ❢♦r k = Z ❛♥❞ Q✳

✹✸

❈♦♥s✐❞❡r t❤❡ ❜✉♥❞❧❡ ˆ πn : E → Mn ✇✐t❤ t❤❡ ✜❜❡r D2✱ s✉❝❤ t❤❛t ∂E = Mn+1✳ ■♥ t❤❡ ❡①❛❝t s❡q✉❡♥❝❡ ♦❢ t❤❡ ♣❛✐r (E, ∂E) t❤❡ ●②③✐♥ ❤♦♠♦♠♦r♣❤✐s♠ ❤❛s t❤❡ ❢♦r♠ jn

q : Hq(Mn) → Hq+2(Mn) : jn q a = [Ωn]a.

❚❤✉s ✇❡ ❣❡t t❤❡ ❡①❛❝t s❡q✉❡♥❝❡ 0 ← kerjn

q−1 ← Hq(Mn+1) ← cokerjn q−2 ← 0.

■♥ t❤❡ ❝❛s❡ ♦❢ r❛t✐♦♥❛❧ ❝♦❡✣❝✐❡♥ts ✇❡ ❣❡t dim Hq(Mn+1) = (dim kerjn

q−1) + (dim cokerjn q−2).

❉❡♥♦t❡ t❤❡ ❇❡tt✐ ♥✉♠❜❡r dim Hq(Mn, Q) ❜② bn

q ✳

❚❤✉s ✇❡ ❤❛✈❡ t❤❡ ❡st✐♠❛t❡ bn+1

q

bn

q−1 + bn q .

✹✹

❉✳ ❱✳ ▼✐❧❧✐♦♥s❤✐❦♦✈ ❤❛s ♦❜t❛✐♥❡❞ r❡s✉❧ts ♦♥ t❤❡ ❇❡tt✐ ♥✉♠❜❡rs bn

q

❢♦r ♠❛♥✐❢♦❧❞s Mn ❞❡✜♥❡❞ ❜② t❤❡ ❣r♦✉♣s Ln✳ ❍✐s ❛♣♣r♦❛❝❤ ✐s ❜❛s❡❞ ♦♥ t❤❡ ❝❛❧❝✉❧❛t✐♦♥s ❜② ▲✳ ●♦♥❝❤❛r♦✈❛ ♦❢ ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ▲✐❡ ❛❧❣❡❜r❛s ❝♦❤♦♠♦❧♦❣✐❡s✳ ❋♦r s✉❝❤ ♠❛♥✐❢♦❧❞s ❤❡ ♣r♦✈❡❞ t❤❛t bn

2 = 3 ❢♦r ❛❧❧ n > 5;

bn

3 = 5 ❢♦r ❛❧❧ n > 11.

✹✺

❉✳ ❱✳ ▼✐❧❧✐♦♥s❤❝❤✐❦♦✈ ✉s❡❞ s♦♠❡ ❝♦♠❜✐♥❛t♦r✐❛❧ ❛r❣✉♠❡♥ts ❛♥❞ t❤❡ ●♦♥❝❤❛r♦✈❛ t❤❡♦r❡♠ t♦ s❦❡t❝❤ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ st❛t❡♠❡♥t bn

q = Fq+2

❢♦r n s✉✣❝✐❡♥t❧② ❧❛r❣❡ ✭n > 3q + 2✮✱ ✇❤❡r❡ Fq+2 ✐s t❤❡ (q + 2)✲t❤ ❋✐❜♦♥❛❝❝✐ ♥✉♠❜❡r✳ ❚❤❛t ✐s Fq+2 = Fq+1 + Fq✱ q 0✱ F0 = 0✱ F1 = 1✳ ❍♦✇❡✈❡r ♥♦ ❞❡t❛✐❧❡❞ ♣r♦♦❢ ♦❢ t❤✐s st❛t❡♠❡♥t ❛♣♣❡❛r❡❞ t✐❧❧ ♥♦✇✳ ❘❡❝❡♥t❧② ❤❡ s✉❣❣❡st❡❞ t♦ ❝♦♥s✐❞❡r t❤❡ ❧❛st st❛t❡♠❡♥t ❛s ❛ ❝♦♥❥❡❝t✉r❡✳ ❯s✐♥❣ t❤❡ ❝♦♠♣✉t❡r✱ ▼✐❧❧✐♦♥s❝❤✐❦♦✈ ❝❛❧❝✉❧❛t❡❞ ❇❡tt✐ ♥✉♠❜❡rs bn

q

❢♦r n 30✳

✹✻

❘❡❢❡r❡♥❝❡s✳ ✶✳ ❇✉❝❤st❛❜❡r ❱✳ ▼✳✱ ❙❤♦❦✉r♦✈ ❆✳ ❱✳✱ ❚❤❡ ▲❛♥❞✇❡❜❡r✕◆♦✈✐❦♦✈ ❛❧❣❡❜r❛ ❛♥❞ ❢♦r♠❛❧ ✈❡❝t♦r ✜❡❧❞s ♦♥ t❤❡ ❧✐♥❡✱ ❋✉♥❝t✳ ❆♥❛❧✳ ❛♥❞ ❆♣♣❧✳✱ ✶✾✼✽✱ ✶✷✱ ◆✸✱ ✶✕✶✶✳ ✷✳ ❇✉❝❤st❛❜❡r ❱✳ ▼✳✱ ❙❡♠✐❣r♦✉♣s ♦❢ ♠❛♣s ✐♥t♦ ❣r♦✉♣s✱ ♦♣❡r❛t♦r ❞♦✉❜❧❡s✱ ❛♥❞ ❝♦♠♣❧❡① ❝♦❜♦r❞✐s♠s✱ ❚♦♣✐❝s ✐♥ t♦♣♦❧♦❣② ❛♥❞ ♠❛t❤❡♠❛t✐❝❛❧ ♣❤②s✐❝s✱ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ❚r❛♥s❧✳ ❙❡r✳ ✷✱ ✶✼✵✱ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ Pr♦✈✐❞❡♥❝❡✱ ❘■✱ ✶✾✾✺✱ ✾✕✸✶ ✸✳ ❇❛❜❡♥❦♦ ■✳ ❑✳✱ ❚❛✐♠❛♥♦✈ ■✳ ❆✳✱ ❖♥ ♥♦♥❢♦r♠❛❧ s✐♠♣❧② ❝♦♥♥❡❝t❡❞ s②♠♣❧❡❝t✐❝ ♠❛♥✐❢♦❧❞s✱ ❙✐❜❡r✐❛♥ ▼❛t❤✳ ❏✳ ✹✶ ✭✷✵✵✵✮✱ ♥♦✳ ✷✱ ✷✵✹✖✷✶✼✳ ✹✳ ❇✉❝❤st❛❜❡r ❱✳ ▼✳✱ ●r♦✉♣s ♦❢ ♣♦❧②♥♦♠✐❛❧ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ ❛ ❧✐♥❡✱ ♥♦♥❢♦r♠❛❧ s②♠♣❧❡❝t✐❝ ♠❛♥✐❢♦❧❞s ❛♥❞ ▲❛♥❞✇❡❜❡r✕◆♦✈✐❦♦✈ ❛❧❣❡❜r❛✱ ❘✉ss✐❛♥ ▼❛t❤✳ ❙✉r✈❡②s✱ ✈✳✺✹✱ ◆ ✹✱ ✶✾✾✾✳

✹✼

✺✳ ❋✉❦s ❉✳ ❇✳✱ ❈♦❤♦♠♦❧♦❣② ♦❢ ✐♥✜♥✐t❡✲❞❡♠❡♥s✐♦♥❛❧ ▲✐❡ ❛❧❣❡❜r❛s✱ ❈♦♥s✉❧t❛♥ts ❇✉r❡❛✉✱ ◆❡✇ ❨♦r❦✱ ✶✾✽✻✳ ✻✳ ●♦♥❝❤❛r♦✈❛ ▲✳ ❱✳✱ ❈♦❤♦♠♦❧♦❣✐❡s ♦❢ ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ❢♦r♠❛❧ ✈❡❝t♦r ✜❡❧❞s ♦♥ t❤❡ str❛✐❣❤t ❧✐♥❡✱ ❋✉♥❝t✳ ❆♥❛❧✳ ❛♥❞ ❆♣♣❧✳✱ ✶✾✼✸✱ ♣❛rt ✶✿ ✼✿✷✱ ✾✶✕✾✼ ♣❛rt ✷✿ ✼✿✸✱ ✶✾✹✕✷✵✸✳ ✼✳ ❏♦❤♥s♦♥ ❋✳ ❊✳ ❆✳✱ ❘❡❡s ❊✳ ●✳✱ ❚❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❣r♦✉♣s ♦❢ ❛❧❣❡❜r❛✐❝ ✈❛r✐❡t✐❡s✱ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ▼❛t❤✳✱ ✶✹✼✹✱ ✶✾✽✾✱ ✼✺✕✽✷✳ ✽✳ ▼✐❧❧✐♦♥s❝❤✐❦♦✈ ❉✳❱✳✱ ❈♦❤♦♠♦❧♦❣② ♦❢ ◆✐❧♠❛♥✐❢♦❧❞s ❛♥❞ ●♦♥t❝❤❛r♦✈❛✬s ❚❤❡♦r❡♠ ✐♥ ✧●❧♦❜❛❧ ❞✐✛❡r❡♥t✐❛❧ ❣❡♦♠❡tr②✿ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ❧❡❣❛❝② ♦❢ ❆❧❢r❡❞ ●r❛②✧✭❇✐❧❜❛♦✱ ✷✵✵✵✮✱ ♣♣✳ ✸✽✶✕✸✽✺✱ ❈♦♥t❡♠♣✳ ▼❛t❤✳✱ ✷✽✽✱ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ Pr♦✈✐❞❡♥❝❡✱ ❘■✱ ✷✵✵✶✳

✹✽

✾✳ ▼❛❧❝❡✈ ❆✳ ■✳✱ ❖♥ ❛ ❝❧❛ss ♦❢ ❤♦♠♦❣❡♥❡♦✉s s♣❛❝❡s✱ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ❚r❛♥s❧❛t✐♦♥ ✶✾✺✶✱ ✭✶✾✺✶✮✳ ♥♦✳ ✸✾✱ ✸✸ ♣♣✳ ✶✵✳ ◆♦♠✐③✉ ❑✳✱ ❖♥ t❤❡ ❝♦❤♦♠♦❧♦❣② ♦❢ ❝♦♠♣❛❝t ❤♦♠♦❣❡♥❡♦✉s s♣❛❝❡s ♦❢ ♥✐❧♣♦t❡♥t ▲✐❡ ❣r♦✉♣s✱ ❆♥♥✳ ♦❢ ▼❛t❤✳ ✭✷✮ ✺✾✱ ✭✶✾✺✹✮✳ ✺✸✶✕✺✸✽✳ ✶✶✳ ❆✉s❧❛♥❞❡r ▲✳✱ ●r❡❡♥ ▲✳✱ ❍❛❤♥ ❋✳✱ ❋❧♦✇s ♦♥ ❤♦♠♦❣❡♥❡♦✉s s♣❛❝❡s✱ ❆♥♥❛❧s ♦❢ ▼❛t❤❡♠❛t✐❝s ❙t✉❞✐❡s ◆✉♠❜❡r ✺✸✳ Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t② Pr❡ss✱ Pr✐♥❝❡t♦♥✱ ◆✳ ❏✳✱ ✶✾✻✸✳ ✶✷✳ ❆✉s❧❛♥❞❡r ▲✳✱ ▲❡❝t✉r❡ ◆♦t❡s ♦♥ ◆✐❧✲❚❤❡t❛ ❢✉♥❝t✐♦♥s✱ ❘❡❣✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ❙❡r✐❡s ✐♥ ▼❛t❤❡♠❛t✐❝s✱ ✸✹✱ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✶✾✼✼✳

✹✾

❆❞❞❡♥❞✉♠✳ ▼❛ss❡② ♣r♦❞✉❝ts✳ ▲❡t (C, d) ✖ ❛ ❞✳❣✳❛✳ ❋♦r a ∈ Cp ♣✉t ¯ a = (−1)pa. ❚❤❡♥ ✇❡ ♦❜t❛✐♥ t❤❡ ✐♥✈♦❧✉t✐♦♥ ♦♥ C✱ ✐✳❡✳ ¯ ab = ¯ a¯ b✱ ¯ ¯ a = a✱ ❛♥❞ dab = (da)b + ¯ a(db) ❢♦r a ∈ Cp. ▲❡t T 0

k = T 0 k (C) ✖ t❤❡ ❛❧❣❡❜r❛ ♦❢ ✉♣♣❡r tr✐❛♥❣✉❧❛r (k×k)✲♠❛tr✐❝❡s

♦✈❡r C ✇✐t❤ ③❡r♦s ♦♥ t❤❡ ❞✐❛❣♦♥❛❧✳ ❋♦r A = (aij) ∈ T 0

k

♣✉t dA = (daij) ❛♥❞ ¯ A = (¯ aij)✳ ▲❡t Jk = (Jk

ij) ∈ T 0 k ✱ s✉❝❤ t❤❛t Jk ij = 0✱ ✐❢ (i, j) = (1, k)✱ ❛♥❞

Jk

1k = 1✳

✺✵

▲❡♠♠❛✳ ▲❡t A = (aij) ∈ T 0

n+1✱ s✉❝❤ t❤❛t ai,i+1 ∈ Cki ❛♥❞

dA = ¯ AA − cJn+1 ❢♦r s♦♠❡ c ∈ C✳ ❚❤❡♥ ✲ dai,i+1 = 0✱ ✲ c ∈ Cm✱ ✇❤❡r❡ m = k1 + · · · + kn − n + 2✱ ✲ dc = 0✳ ❙❤♦✇ t❤❛t dc = 0✳ ❲❡ ❤❛✈❡✿ d ¯ A = − ¯ dA = −A ¯ A − cJn+1. ❯s✐♥❣ t❤❛t Jn+1A = AJn+1 = 0✱ ✇❡ ♦❜t❛✐♥ d ¯ AA = (d ¯ A)A + ¯ ¯ A(dA) = −A ¯ AA + A ¯ AA = 0. ❙♦ (dc)Jn+1 = d( ¯ AA) − ddA = 0.

✺✶

❉❡✜♥✐t✐♦♥✳ ❚❛❦❡ n ❤♦♠♦❣❡♥❡♦✉s ❡❧❡♠❡♥ts a1, . . . , an ✐♥ C✱ ✇❤✐❝❤ ❛r❡ ❝♦❝②❝❧❡s✱ ✐✳❡✳ dai = 0✱ i = 1, . . . , n✳ ❆ss✉♠❡ t❤❛t t❤❡r❡ ❡①✐sts ❛ ♠❛tr✐① A ∈ T 0

n+1 s✉❝❤ t❤❛t✿

✲ ai,i+1 = ai ✲ A s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥ dA = ¯ AA − cJn+1 ❢♦r s♦♠❡ c ∈ C✳ ■♥ t❤✐s ❝❛s❡ ✐t ✐s t♦❧❞ t❤❛t t❤❡ ▼❛ss❡② ♣r♦❞✉❝t a1, . . . , an ♦❢ t❤❡ ❝♦❝②❝❧❡s a1, . . . , an ✐s ❞❡✜♥❡❞ ❛♥❞ ❡q✉❛❧s ❝♦❝②❝❧❡ c✳

✺✷

❊①❛♠♣❧❡s n = 2✳

  

0 ¯ a1 ¯ a13 ¯ a2

     

a1 a13 a2

   =   

0 ¯ a1a2

   .

❙♦ c

  

1

   =   

0 ¯ a1a2

   −   

da1 da13 da2

  

❛♥❞ da1 = da2 = 0, c = a1, a2 = ¯ a1a2 − da13 ❢♦r s♦♠❡ a13✳

✺✸

❊①❛♠♣❧❡s n = 3✳

    

0 ¯ a1 ¯ a13 ¯ a14 ¯ a2 ¯ a24 ¯ a3

         

a1 a13 a14 a2 a24 a3

     =     

0 ¯ a1a2 ¯ a1a24 + ¯ a13a3 ¯ a2a3

    

❙♦ c

    

1

     =     

0 ¯ a1a2 ¯ a1a24 + ¯ a13a3 ¯ a2a3

    −     

da1 da13 da14 da2 da24 da3

     .

❙♦✱ dai = 0✱ i = 1, 2, 3✱ ¯ a1a2 = da13✱ ¯ a2a3 = da24✱ ❛♥❞ a1, a2, a3 = c = ¯ a1a24 + ¯ a13a3✱ deg c = k1 + k2 + k3 − 1✳

✺✹

❊①❛♠♣❧❡s n = 3✳ H∗(M3) = H∗(Λ(ω1, ω2, ω3), d)✱ ✇❤❡r❡ dω1 = dω2 = 0✱ dω3 = ω2 ∧ ω1✳ ❚❤❡ ❣❡♥❡r❛t♦rs ♦❢ H∗(M3)✿ [ω1], [ω2], [ω3 ∧ ω1], [ω3 ∧ ω2], [ω3 ∧ ω2 ∧ ω1]. ❙♦✱ ❢♦r a1 = ω1, a2 = ω2, a3 = ω1 ⇒ a13 = ω3, a24 = −ω3 ❛♥❞ ω1, ω2, ω1 = −2ω3 ∧ ω1✱ a1 = ω1, a2 = ω1, a3 = ω2 ⇒ a13 = 0, a24 = ω3 ❛♥❞ ω1, ω1, ω2 = ω3 ∧ ω1✱ a1 = ω1, a2 = ω2, a3 = ω2 ⇒ a13 = ω3, a24 = 0 ❛♥❞ ω1, ω2, ω2 = −ω3 ∧ ω2✳

✺✺

❚❤❡ ♠❛tr✐① ❡q✉❛t✐♦♥ dA = ¯ AA − cJn+1 ❢♦r n 4 ❣✐✈❡s t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥s✿ dai,i+1 = dai = 0, dai,k =

k−1

• q=i+1

¯ ai,qaq,k, i + 2 k n ❛♥❞ a1, . . . , an = c =

n

• q=2

¯ a1,qaq,n+1 − da1,n+1, ✇❤❡r❡ dc = 0✳ ❙♦ dai,i+2 = ¯ ai,i+1ai+1,i+2 = ¯ aiai+1, dai,i+3 = ¯ ai,i+1ai+1,i+3 + ¯ ai,i+2ai+2,i+3 = ai, ai+1, ai+2.

✺✻

❊①❛♠♣❧❡ n = 4✳ a1 a2 a3 a13 a24 a1, a2, a3 ω1 ω2 ω1 ω3 −ω3 2ω3 ∧ ω1 = dω4 ω2 ω1 ω1 −ω3 ω3 ∧ ω1 = 1

2dω4

ω1 ω2 ω2 ω3 ω3 ∧ ω2

✺✼

❲❡ ❤❛✈❡ a1, a2, a3, a4 = ¯ a12a25 + ¯ a13a35 + ¯ a14a45. ❋♦r ω2, ω1, ω1, ω1✿ da25 = ω1, ω1, ω1 = 0 ⇒ a25 = 0 da35 = −ω1 ∧ ω1 = 0 ⇒ a35 = 0 da14 = ω2, ω1, ω1 = 1 2dω4 ⇒ a14 = 1 2ω4. ❙♦✱ ✇❡ ♦❜t❛✐♥❡❞✿ ω2, ω1, ω1, ω1 = −1 2ω4 ∧ ω1 = 0.

✺✽

❋♦r ω1, ω2, ω2, ω2✿ a25 = 0, a35 = 0 ❛♥❞ da14 = ω1, ω2, ω2 = ω3 ∧ ω2. ❲❡ ❝❛♥✬t ✜♥❞ s✉❝❤ a14 ❛♥❞ t❤❡r❡❢♦r❡ t❤❡ ▼❛ss❡② ♣r♦❞✉❝t ω1, ω2, ω2, ω2 ✐s ♥♦t ✇❡❧❧ ❞❡✜♥❡❞ ✐♥ H∗(M4)✳

✺✾

■♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❛❧❣❡❜r❛ ♦❢ ✈❡❝t♦r ✜❡❧❞s ♦❢ t❤❡ ❧✐♥❡✳ ■♥tr♦❞✉❝❡✿ t❤❡ ❣r♦✉♣ L∞ = lim← Ln ▲✐❡ ❛❧❣❡❜r❛ L∞ = lim← Ln ❛♥❞ ❛❧❣❡❜r❛ ♦❢ ♦♣❡r❛t♦rs A∞ = lim← An✳ ▲❡t l1 = {xk+1 d

dx, k 1} ❜❡ t❤❡ ✇❡❧❧ ❦♥♦✇♥ ▲✐❡ ❛❧❣❡❜r❛ ♦❢ ✈❡❝t♦r

✜❡❧❞s ♦♥ t❤❡ ❧✐♥❡✳ ❲❡ ❤❛✈❡ L∞ ∼ = l1✳

✻✵

❚❤❡♦r❡♠✳✭▲✳ ❱✳ ●♦♥❝❤❛r♦✈❛✱ ✶✾✼✸✮ dim Hq

k(l1) =

• 1, ✐❢ k = 3q2±q

2

, 0, ♦t❤❡r✇✐s❡ ❚❤✉s✱ dim Hq(l1) = 2 ❢♦r q 1✳ ❚❤❡ ❝♦❤♦♠♦❧♦❣✐❝❛❧ ♣r♦❞✉❝t ✐♥ H∗(l1) ✐s tr✐✈✐❛❧✳ ■t ✇❛s ❱✳ ▼✳ ❇✉❝❤st❛❜❡r ✭✶✾✼✽✮ ✇❤♦ r❛✐s❡❞ t❤❡ ♣r♦❜❧❡♠ ✇❤❡t❤❡r H∗(l1) ✐s ❣❡♥❡r❛t❡❞✱ ✇✐t❤ r❡s♣❡❝t t♦ ▼❛ss❡② ♣r♦❞✉❝ts✱ ❜② H1(l1)✳

✻✶

❚❤❡ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣✳ ▲❡t ✉s ✜① ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ Rn = Rk × Rl✱ n = k + l✱ ❛♥❞ ❛ ❜✐❧✐♥❡❛r ♠❛♣♣✐♥❣ B : Rk × Rk → Rl✳ ❲❡ ❞❡✜♥❡ ❛ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦♥ Rn ❜② t❤❡ ❢♦r♠✉❧❛ (v1, w1) · (v2, w2) = (v1 + v2, w1 + w2 + B(v1, v2)) ✇❤❡r❡ vi ∈ Rk✱ wi ∈ Rl✱ i = 1, 2✳ ◆♦t❡ t❤❡ r❡❧❛t✐♦♥ B(v1, v2) = A(v1)v2, ✇❤❡r❡ A ✐s t❤❡ ❧✐♥❡❛r ♠❛♣♣✐♥❣ Rk → Hom(Rk, Rl)✳ ❚❤✉s ✇❡ ♦❜t❛✐♥ ❛ ❣r♦✉♣ str✉❝t✉r❡ ♦♥ Rn✱ ✇❤✐❝❤ ✐s ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❢♦r ♥♦♥s②♠♠❡tr✐❝ ♠❛♣♣✐♥❣ B✳

✻✷

❆ ❧✐♥❡❛r ❝❤❛♥❣❡ ♦❢ ❝♦♦r❞✐♥❛t❡s B = (B1, B2) ∈ GL(k, R) × GL(l, R) ⊂ GL(n, R) ❣✐✈❡s ❛ ♥❡✇ ♠✉❧t✐♣❧✐❝❛t✐♦♥ (v1, w1) ∗ (v2, w2) = (v1 + v2, w1 + w2 + B−1

2 B(B1v1, B1v2)).

❋♦r t❤❡ s❝❛❧❛r ♠❛tr✐① τE ✇❡ ❣❡t (v1, w1) ∗τ (v2, w2) = (v1 + v2, w1 + w2 + τB(v1, v2)) ❛♥❞ t❤✐s ❣✐✈❡s ❛ ❞❡❢♦r♠❛t✐♦♥ ✐♥t♦ t❤❡ st❛♥❞❛r❞ ❛❞❞✐t✐♦♥✳ ◆♦t❡✿ t❤✐s ✐s ❛ ❜✐❧✐♥❡❛r ❞❡❢♦r♠❛t✐♦♥✳

✻✸

❚♦ ♦❜t❛✐♥ t❤❡ ✇❡❧❧✲❦♥♦✇♥ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣ t❛❦❡ k = 2✱ l = 1 ❛♥❞ ❢♦r vi = (xi, yi) ♣✉t B : R2 × R2 → R1 : B(v1, v2) =

• x1

y1 τ x2 y2

• = τx1y2.

❚❤❡ ❍❡✐s❡♥❜❡r❣ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦♥ R3✿ (x1, y1, w1) · (x2, y2, w2) = (x1 + x2, y1 + y2, w1 + w2 + τx1y2).

✻✹