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❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s

✐♥ ❝♦❧❧❛❜♦r❛t✐♦♥ ✇✐t❤ ▼✐❝❤❛❡❧ ▲❡s♥✐❝❦ ❘♦② ❩❤❛♦ ❆✉❣✉st ✶✺✱ ✷✵✶✽

❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✶ ✴ ✶✼

❚❛❜❧❡ ♦❢ ❈♦♥t❡♥ts

✶

❇❛❝❦❣r♦✉♥❞ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥ ❉✐✣❝✉❧t✐❡s

✷

❈♦♠♣✉t✐♥❣ ❇✐✜❧tr❛t✐♦♥s

• r❛❞❡s ♦❢ ❆♣♣❡❛r❛♥❝❡

❘❡s♦❧✈✐♥❣ ❘❡❧❛t✐♦♥s

✸

❈♦♠♠❡♥ts

❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✷ ✴ ✶✼

❉❡✜♥✐t✐♦♥s

❉❡✜♥❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛rt✐❛❧ ♦r❞❡r ♦♥ N✷✿ (x✶, y✶) ≤ (x✷, y✷) ✐✛ x✶ ≥ x✷ ❛♥❞ y✶ ≤ y✷✳ ❆ ❜✐✜❧tr❛t✐♦♥ B ✐s ❛ s❡t ♦❢ ✈❡❝t♦r s♣❛❝❡s {BP}P∈N✷ s✉❝❤ t❤❛t BP ⊂ BQ ❢♦r ❛❧❧ P ≤ Q✳

• ✐✈❡♥ ❛ s❡t ♦❢ ♣♦✐♥ts S ❛♥❞ ♥♦♥✲♥❡❣❛t✐✈❡ r❡❛❧ ♥✉♠❜❡r t ∈ N✱ t❤❡ t s❦❡❧❡t♦♥

♦❢ S ✐s t❤❡ ❣r❛♣❤ ✇❤♦s❡ ✈❡rt✐❝❡s ❛r❡ t❤❡ ♣♦✐♥ts ✐♥ S ❛♥❞ ❛♥❞ ❡❞❣❡ ❡①✐sts ❜❡t✇❡❡♥ t✇♦ ✈❡rt✐❝❡s ✐❢ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ♣♦✐♥ts ✐s ≤ t✳ ■♥ t❤❡ ❞❡❣r❡❡✲❘✐♣s ❜✐✜❧tr❛t✐♦♥ ♦❢ ❛ s❡t ♦❢ ♣♦✐♥ts S✱ ❛ ✵✲s✐♠♣❧❡① P ❡①✐sts ❛t ❛ ♣♦✐♥t (d, t) ✐❢ ✐♥ t❤❡ t s❦❡❧❡t♦♥ ♦❢ S✱ ✇❡ ❤❛✈❡ ❞❡❣(P) ≥ d✳ ❆ s✐♠♣❧❡① σ ❡①✐sts ❛t ❛ ♣♦✐♥t ✐❢ ❛❧❧ ♦❢ ✐ts ♣♦✐♥ts ❛♥❞ ❛❧❧ ♦❢ ✐ts ❡❞❣❡s ❡①✐st✳

❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✸ ✴ ✶✼

❊①❛♠♣❧❡

✷ ✷ ✶

❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✹ ✴ ✶✼

■♥❝♦♠♣❛r❛❜❧❡ ●r❛❞❡s

❲❤❛t ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♣♦✐♥ts s✉❝❤ t❤❛t ♣♦✐♥t A ❛♣♣❡❛rs❄

Pr♦♣♦s✐t✐♦♥

❋♦r ❛♥② s✐♠♣❧❡① ✐♥ ❛ ❞❡❣r❡❡✲❘✐♣s ❜✐✜❧tr❛t✐♦♥ ♦❢ ❛ ✜♥✐t❡ s❡t ♦❢ ♣♦✐♥ts ✱ t❤❡r❡ ❡①✐sts ❛ ✜♥✐t❡ s❡t ♦❢ ♣♦✐♥ts s✉❝❤ t❤❛t ❡①✐sts ❛t ❛ ♣♦✐♥t ✐✛ s✉❝❤ t❤❛t ✳ ❚❤❡ ♣♦✐♥ts ✐♥ ❛ ♠✐♥✐♠❛❧ s✉❝❤ ❛r❡ ❝❛❧❧❡❞ t❤❡ ❣r❛❞❡s ♦❢ ❛♣♣❡❛r❛♥❝❡ ♦❢ ✳

❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✺ ✴ ✶✼

■♥❝♦♠♣❛r❛❜❧❡ ●r❛❞❡s

❲❤❛t ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♣♦✐♥ts s✉❝❤ t❤❛t ♣♦✐♥t A ❛♣♣❡❛rs❄

Pr♦♣♦s✐t✐♦♥

❋♦r ❛♥② s✐♠♣❧❡① σ ✐♥ ❛ ❞❡❣r❡❡✲❘✐♣s ❜✐✜❧tr❛t✐♦♥ ♦❢ ❛ ✜♥✐t❡ s❡t ♦❢ ♣♦✐♥ts S✱ t❤❡r❡ ❡①✐sts ❛ ✜♥✐t❡ s❡t ♦❢ ♣♦✐♥ts Σ s✉❝❤ t❤❛t σ ❡①✐sts ❛t ❛ ♣♦✐♥t Q ✐✛ ∃P ∈ Σ s✉❝❤ t❤❛t P ≤ Q✳ ❚❤❡ ♣♦✐♥ts ✐♥ ❛ ♠✐♥✐♠❛❧ s✉❝❤ Σ ❛r❡ ❝❛❧❧❡❞ t❤❡ ❣r❛❞❡s ♦❢ ❛♣♣❡❛r❛♥❝❡ ♦❢ σ✳

❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✺ ✴ ✶✼

❚❛❜❧❡ ♦❢ ❈♦♥t❡♥ts

✶

❇❛❝❦❣r♦✉♥❞ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥ ❉✐✣❝✉❧t✐❡s

✷

❈♦♠♣✉t✐♥❣ ❇✐✜❧tr❛t✐♦♥s

• r❛❞❡s ♦❢ ❆♣♣❡❛r❛♥❝❡

❘❡s♦❧✈✐♥❣ ❘❡❧❛t✐♦♥s

✸

❈♦♠♠❡♥ts

❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✻ ✴ ✶✼

❙t❡♣ ✶✿ ❉❡❛❧ ✇✐t❤ ✵✲❙✐♠♣❧✐❝❡s

❢✉♥❝t✐♦♥ ●❡♥❡r❛t❡❱❡rt❡①▼✉❧t✐❣r❛❞❡s✭d, k✮ dk ← {d(k, j) : j = k} Gradesk ← {(✵, ✵)} sort(dk) ❢♦r i ← ✵; i < size(dk); i + + ❞♦ ✇❤✐❧❡ i + ✶ < size(dk) ❛♥❞ dk[i + ✶] = dk[i] ❞♦ i + + ❡♥❞ ✇❤✐❧❡ Gradesk ← Gradesk ∪ {(i + ✶, dk(i))} ❡♥❞ ❢♦r ❡♥❞ ❢✉♥❝t✐♦♥

❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✼ ✴ ✶✼

❙t❡♣ ✷✿ ■♥❞✉❝t t♦ n✲❙✐♠♣❧✐❝❡s

Pr♦♣♦s✐t✐♦♥

▲❡t σ ❜❡ ❛ n✲s✐♠♣❧❡① ✇✐t❤ n ≥ ✶✱ τ ❜❡ ❛ ❢❛❝❡ ♦❢ σ ❛♥❞ P = σ\τ✳ ▲❡t D = ♠❛①Q∈τ d(P, Q)✳ ▲❡t Sσ, Sτ, SP ❜❡ t❤❡ s❡t ♦❢ ♣♦✐♥ts ✇❤❡r❡ σ, τ, P ❡①✐st r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡♥ Sσ = Sτ ∩ SP ∩ {(x, y) : y ≥ D}✳ ❙✇❡❡♣✐♥❣ ❧✐♥❡ ❢r♦♠ ❧❡❢t t♦ r✐❣❤t✳ ❋♦r ❡❛❝❤ ✱ ♠❛✐♥t❛✐♥ ♠❛① ♠✐♥ ♠✐♥ ✳ ❚❤✐s ♠❛①✐♠✉♠ ❝❤❛♥❣❡s ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✇❡ ❤❛✈❡ ❡♥❝♦✉♥t❡r❡❞ ❛ ❣r❛❞❡ ♦❢ ❛♣♣❡❛r❛♥❝❡ ❢♦r ✳

❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✽ ✴ ✶✼

❙t❡♣ ✷✿ ■♥❞✉❝t t♦ n✲❙✐♠♣❧✐❝❡s

Pr♦♣♦s✐t✐♦♥

▲❡t σ ❜❡ ❛ n✲s✐♠♣❧❡① ✇✐t❤ n ≥ ✶✱ τ ❜❡ ❛ ❢❛❝❡ ♦❢ σ ❛♥❞ P = σ\τ✳ ▲❡t D = ♠❛①Q∈τ d(P, Q)✳ ▲❡t Sσ, Sτ, SP ❜❡ t❤❡ s❡t ♦❢ ♣♦✐♥ts ✇❤❡r❡ σ, τ, P ❡①✐st r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡♥ Sσ = Sτ ∩ SP ∩ {(x, y) : y ≥ D}✳ ❙✇❡❡♣✐♥❣ ❧✐♥❡ ❢r♦♠ ❧❡❢t t♦ r✐❣❤t✳ ❋♦r ❡❛❝❤ x ∈ N✱ ♠❛✐♥t❛✐♥ ♠❛①(D, ♠✐♥(x,y)∈SP y, ♠✐♥(x,y)∈Sτ y)✳ ❚❤✐s ♠❛①✐♠✉♠ ❝❤❛♥❣❡s ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✇❡ ❤❛✈❡ ❡♥❝♦✉♥t❡r❡❞ ❛ ❣r❛❞❡ ♦❢ ❛♣♣❡❛r❛♥❝❡ ❢♦r σ✳

❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✽ ✴ ✶✼

Ps❡✉❞♦✲❝♦❞❡

❢✉♥❝t✐♦♥ ❈♦♠❜✐♥❡▼✉❧t✐❣r❛❞❡s✭SP, Sτ, D✮ G ← {} i✶ ← ✵, i✷ ← ✵ d = ♠❛①(D, SP[i✶].y, Sτ[i✷].y) ✇❤✐❧❡ i✶ < |SP| ❛♥❞ i✷ < |Sτ| ❞♦ minX ← ♠✐♥(SP[i✶].x, Sτ[i✷].x) ✐❢ SP[i✶].x == minX t❤❡♥ i✶ + + ❡♥❞ ✐❢ ✐❢ Sτ[i✷].x == minX t❤❡♥ i✷ + + ❡♥❞ ✐❢ d′ = ♠❛①(D, SP[i✶].y, Sτ[i✷].y) ✐❢ d′ > d t❤❡♥ G ← G ∪ {(minX, d)} d ← d′ ❡♥❞ ✐❢ ❡♥❞ ✇❤✐❧❡ ❡♥❞ ❢✉♥❝t✐♦♥

❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✾ ✴ ✶✼

▼❛t❤❡♠❛t✐❝❛❧ ❇❛❝❦❣r♦✉♥❞

• ✐✈❡♥ ❛ ❜✐✜❧tr❛t✐♦♥ B✱ ✇❡ ❣❡t ❛ ❝❤❛✐♥ ❝♦♠♣❧❡① ♦❢ ✷✲❉ ❜✐♣❡rs✐st❡♥❝❡ ♠♦❞✉❧❡s

Cj+✶

dj+✶

→ Cj

dj

→ Cj−✶

dj−✶

→ · · · d✶ → C✵ → ✵ ❛♥❞ Hj(B) ∼ = ❦❡r dj/ ✐♠ dj+✶. ❘■❱❊❚ r❡q✉✐r❡s t❤❡ Ci t♦ ❜❡ ❢r❡❡ ♦r ♦♥❡✲❝r✐t✐❝❛❧✳

❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✶✵ ✴ ✶✼

▼✐♥✐♠❛❧ Pr❡s❡♥t❛t✐♦♥

▲❡t M[P] ❜❡ t❤❡ ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ♠♦❞✉❧❡ ✇✐t❤ M[P]Q =

• k

✐❢ Q ≥ P ✵ ♦t❤❡r✇✐s❡✳ ❋♦r ❛♥② s❡t ♦❢ ♣♦✐♥ts Σ✱ ❧❡t F(Σ) =

P∈Σ M[P]✳

Pr♦♣♦s✐t✐♦♥

❋♦r ❛ ❜✐♣❡rs✐st❡♥❝❡ ♠♦❞✉❧❡ N ♦❢ r❛♥❦ ✶ ✇✐t❤ ❣r❛❞❡s ♦❢ ❛♣♣❡❛r❛♥❝❡ ❣✐✈❡♥ ❜② t❤❡ s❡t G✱ ❧❡t R ❜❡ t❤❡ s❡t ♦❢ ♣♦✐♥ts ✇❤✐❝❤ ❛r❡ ❧❡❛st ✉♣♣❡r ❜♦✉♥❞s ♦❢ ♣❛✐rs ♦❢ ❞✐st✐♥❝t ♣♦✐♥ts ✐♥ G✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ s✉r❥❡❝t✐✈❡ ♠❛♣ F[G] → N ❛♥❞ ❛♥ ✐♥❥❡❝t✐♦♥ F[R] ֒ → F[G] ✇❤♦s❡ ✐♠❛❣❡ ✐s t❤❡ ❦❡r♥❡❧ ♦❢ t❤❡ ❢♦r♠❡r ♠❛♣✳

❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✶✶ ✴ ✶✼

❊①❛♠♣❧❡

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ · · · k✷ k✷ k · · · k✷ k ✵ · · · k ✵ ✵ → ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ · · · k k k · · · k k ✵ · · · k ✵ ✵

❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✶✷ ✴ ✶✼

❊①❛♠♣❧❡

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ · · · k✷ k ✵ · · · k ✵ ✵ · · · ✵ ✵ ✵ → ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ · · · k✷ k✷ k · · · k✷ k ✵ · · · k ✵ ✵

❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✶✸ ✴ ✶✼

❘❡s♦❧✈✐♥❣ dj+✶

■♥ ❣❡♥❡r❛❧✱ Ci =

i✲s✐♠♣❧✐❝❡s σ C[σ] ✇❤❡r❡ C[σ]P ✐s k ✐❢ σ ❡①✐sts ❛t P ❛♥❞ ✵

♦t❤❡r✇✐s❡✳ ▲❡t Gi ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ t❤❡ ❣r❛❞❡s ♦❢ ❛♣♣❡❛r❛♥❝❡ ♦❢ ❛❧❧ t❤❡ i s✐♠♣❧✐❝❡s✱ ❝♦✉♥t❡❞ ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t②✱ ❛♥❞ ❧❡t Ri ❜❡ t❤❡ s❡t ♦❢ ❧❡❛st ✉♣♣❡r ❜♦✉♥❞s ♦❢ ♣❛✐rs ♦❢ ❞✐st✐♥❝t ♣♦✐♥ts ❝♦♠✐♥❣ ❢r♦♠ t❤❡ s❛♠❡ s✐♠♣❧❡① ✐♥ Gi✳ ❚❤❡♥ ✇❡ ❤❛✈❡ t✇♦ ♠❛♣s F[Gj+✶] → Cj+✶ → Cj ❛♥❞ F[Rj] → F[Gj]. ❈❧❛✐♠✿ ❲❡ ❝❛♥ ❧✐❢t t❤❡ ✜rst ♠❛♣ t♦ ❛ ♠❛♣

✶

✳ ❚❤✐s ❣✐✈❡s ❛ ♠❛♣✿

✶

❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✶✹ ✴ ✶✼

❘❡s♦❧✈✐♥❣ dj+✶

■♥ ❣❡♥❡r❛❧✱ Ci =

i✲s✐♠♣❧✐❝❡s σ C[σ] ✇❤❡r❡ C[σ]P ✐s k ✐❢ σ ❡①✐sts ❛t P ❛♥❞ ✵

♦t❤❡r✇✐s❡✳ ▲❡t Gi ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ t❤❡ ❣r❛❞❡s ♦❢ ❛♣♣❡❛r❛♥❝❡ ♦❢ ❛❧❧ t❤❡ i s✐♠♣❧✐❝❡s✱ ❝♦✉♥t❡❞ ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t②✱ ❛♥❞ ❧❡t Ri ❜❡ t❤❡ s❡t ♦❢ ❧❡❛st ✉♣♣❡r ❜♦✉♥❞s ♦❢ ♣❛✐rs ♦❢ ❞✐st✐♥❝t ♣♦✐♥ts ❝♦♠✐♥❣ ❢r♦♠ t❤❡ s❛♠❡ s✐♠♣❧❡① ✐♥ Gi✳ ❚❤❡♥ ✇❡ ❤❛✈❡ t✇♦ ♠❛♣s F[Gj+✶] → Cj+✶ → Cj ❛♥❞ F[Rj] → F[Gj]. ❈❧❛✐♠✿ ❲❡ ❝❛♥ ❧✐❢t t❤❡ ✜rst ♠❛♣ t♦ ❛ ♠❛♣ F[Gj+✶] → F[Gj]✳ ❚❤✐s ❣✐✈❡s ❛ ♠❛♣✿

✶

❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✶✹ ✴ ✶✼

❘❡s♦❧✈✐♥❣ dj+✶

■♥ ❣❡♥❡r❛❧✱ Ci =

i✲s✐♠♣❧✐❝❡s σ C[σ] ✇❤❡r❡ C[σ]P ✐s k ✐❢ σ ❡①✐sts ❛t P ❛♥❞ ✵

♦t❤❡r✇✐s❡✳ ▲❡t Gi ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ t❤❡ ❣r❛❞❡s ♦❢ ❛♣♣❡❛r❛♥❝❡ ♦❢ ❛❧❧ t❤❡ i s✐♠♣❧✐❝❡s✱ ❝♦✉♥t❡❞ ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t②✱ ❛♥❞ ❧❡t Ri ❜❡ t❤❡ s❡t ♦❢ ❧❡❛st ✉♣♣❡r ❜♦✉♥❞s ♦❢ ♣❛✐rs ♦❢ ❞✐st✐♥❝t ♣♦✐♥ts ❝♦♠✐♥❣ ❢r♦♠ t❤❡ s❛♠❡ s✐♠♣❧❡① ✐♥ Gi✳ ❚❤❡♥ ✇❡ ❤❛✈❡ t✇♦ ♠❛♣s F[Gj+✶] → Cj+✶ → Cj ❛♥❞ F[Rj] → F[Gj]. ❈❧❛✐♠✿ ❲❡ ❝❛♥ ❧✐❢t t❤❡ ✜rst ♠❛♣ t♦ ❛ ♠❛♣ F[Gj+✶] → F[Gj]✳ ❚❤✐s ❣✐✈❡s ❛ ♠❛♣✿ f : F[Gj+✶] ⊕ F[Rj] → F[Gj].

❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✶✹ ✴ ✶✼

❘❡s♦❧✈✐♥❣ dj

❋♦r ❡❛❝❤ j − ✶ s✐♠♣❧❡① τ✱ ❧❡t L[τ] ❜❡ t❤❡ ❣r❡❛t❡st ✉♣♣❡r ❜♦✉♥❞ ♦❢ t❤❡ ❣r❛❞❡s ♦❢ ❛♣♣❡❛r❛♥❝❡ ♦❢ τ ❛♥❞ ❧❡t L ❜❡ t❤❡ s❡t ❝♦♥s✐st✐♥❣ ♦❢ L[τ] ❢♦r ❛❧❧ s✉❝❤ τ✳ ❚❤❡♥✱ t❤❡r❡ ❡①✐sts ❛ ❝❛♥♦♥✐❝❛❧ ✐♥❥❡❝t✐♦♥ Cj−✶ ֒ → F[L]✳ ❈♦♠♣♦s✐♥❣ ✇✐t❤ ❣✐✈❡s ✉s ❛ ♠❛♣

✶

❛♥❞ t❤✉s ❛ ♠❛♣

Pr♦♣♦s✐t✐♦♥

■♥ t❤❡ ♣r❡s❡♥t❛t✐♦♥

✶

✱ ❦❡r ✐♠

✶

❦❡r ✐♠ ✳

❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✶✺ ✴ ✶✼

❘❡s♦❧✈✐♥❣ dj

❋♦r ❡❛❝❤ j − ✶ s✐♠♣❧❡① τ✱ ❧❡t L[τ] ❜❡ t❤❡ ❣r❡❛t❡st ✉♣♣❡r ❜♦✉♥❞ ♦❢ t❤❡ ❣r❛❞❡s ♦❢ ❛♣♣❡❛r❛♥❝❡ ♦❢ τ ❛♥❞ ❧❡t L ❜❡ t❤❡ s❡t ❝♦♥s✐st✐♥❣ ♦❢ L[τ] ❢♦r ❛❧❧ s✉❝❤ τ✳ ❚❤❡♥✱ t❤❡r❡ ❡①✐sts ❛ ❝❛♥♦♥✐❝❛❧ ✐♥❥❡❝t✐♦♥ Cj−✶ ֒ → F[L]✳ ❈♦♠♣♦s✐♥❣ ✇✐t❤ Cj ❣✐✈❡s ✉s ❛ ♠❛♣ Cj → Cj−✶ → F[L] ❛♥❞ t❤✉s ❛ ♠❛♣ g : F[Gj] → F[L].

Pr♦♣♦s✐t✐♦♥

■♥ t❤❡ ♣r❡s❡♥t❛t✐♦♥

✶

✱ ❦❡r ✐♠

✶

❦❡r ✐♠ ✳

❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✶✺ ✴ ✶✼

❘❡s♦❧✈✐♥❣ dj

❋♦r ❡❛❝❤ j − ✶ s✐♠♣❧❡① τ✱ ❧❡t L[τ] ❜❡ t❤❡ ❣r❡❛t❡st ✉♣♣❡r ❜♦✉♥❞ ♦❢ t❤❡ ❣r❛❞❡s ♦❢ ❛♣♣❡❛r❛♥❝❡ ♦❢ τ ❛♥❞ ❧❡t L ❜❡ t❤❡ s❡t ❝♦♥s✐st✐♥❣ ♦❢ L[τ] ❢♦r ❛❧❧ s✉❝❤ τ✳ ❚❤❡♥✱ t❤❡r❡ ❡①✐sts ❛ ❝❛♥♦♥✐❝❛❧ ✐♥❥❡❝t✐♦♥ Cj−✶ ֒ → F[L]✳ ❈♦♠♣♦s✐♥❣ ✇✐t❤ Cj ❣✐✈❡s ✉s ❛ ♠❛♣ Cj → Cj−✶ → F[L] ❛♥❞ t❤✉s ❛ ♠❛♣ g : F[Gj] → F[L].

Pr♦♣♦s✐t✐♦♥

■♥ t❤❡ ♣r❡s❡♥t❛t✐♦♥ F[Gj+✶] ⊕ F[Rj] f → F[Gj]

g

→ F[L]✱ Hj(B) ∼ = ❦❡r dj/ ✐♠ dj+✶ ∼ = ❦❡r g/ ✐♠ f ✳

❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✶✺ ✴ ✶✼

❚❛❜❧❡ ♦❢ ❈♦♥t❡♥ts

✶

❇❛❝❦❣r♦✉♥❞ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥ ❉✐✣❝✉❧t✐❡s

✷

❈♦♠♣✉t✐♥❣ ❇✐✜❧tr❛t✐♦♥s

• r❛❞❡s ♦❢ ❆♣♣❡❛r❛♥❝❡

❘❡s♦❧✈✐♥❣ ❘❡❧❛t✐♦♥s

✸

❈♦♠♠❡♥ts

❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✶✻ ✴ ✶✼

❋✉t✉r❡ ❲♦r❦

P❡r❢♦r♠❛♥❝❡ s♣❡❡❞✉♣s ❈❧❡❛r✐♥❣ ❉✉❛❧✐③❛t✐♦♥

❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✶✼ ✴ ✶✼