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# Simultaneously vanishing higher derived limits

August 08, 2019

## Transcript

1. ### Simultaneously vanishing higher derived limits (Joint work with Jeffrey Bergfalk)

Chris Lambie-Hanson Department of Mathematics and Applied Mathematics Virginia Commonwealth University Oaxaca 8 August 2019

3. ### Additivity Deﬁnition A homology theory is additive on a class

of topological spaces C if, for every natural number p and every family {Xi | i ∈ J} such that each Xi and J Xi are in C, we have J Hp(Xi ) ∼ = Hp( J Xi ) via the map induced by the inclusions Xi → J Xi .
4. ### Additivity of strong homology Let Xn denote the n-dimensional Hawaiian

earring, i.e., the one-point compactiﬁcation of an inﬁnite countable sum of copies of the n-dimensional open unit ball. Let ¯ Hp(X) denote the pth strong homology group of X.
5. ### Additivity of strong homology Let Xn denote the n-dimensional Hawaiian

earring, i.e., the one-point compactiﬁcation of an inﬁnite countable sum of copies of the n-dimensional open unit ball. Let ¯ Hp(X) denote the pth strong homology group of X. Theorem (Mardeˇ si´ c-Prasolov, ‘88) Suppose that 0 < p < n are natural numbers. Then N ¯ Hp(Xn) = ¯ Hp( N Xn) if and only if limn−p A = 0.
6. ### Additivity of strong homology Let Xn denote the n-dimensional Hawaiian

earring, i.e., the one-point compactiﬁcation of an inﬁnite countable sum of copies of the n-dimensional open unit ball. Let ¯ Hp(X) denote the pth strong homology group of X. Theorem (Mardeˇ si´ c-Prasolov, ‘88) Suppose that 0 < p < n are natural numbers. Then N ¯ Hp(Xn) = ¯ Hp( N Xn) if and only if limn−p A = 0. Consequently, if strong homology is additive on closed subsets of Euclidean space, then limn A = 0 for all n ≥ 1.

8. ### Compact supports Deﬁnition A homology theory has compact supports on

a class of topological spaces C if, for every natural number p and every space X in C, Hp(X) is isomorphic to the direct limit of Hp(K) for compact subspaces K of X via the map induced by the inclusion maps K → X.
9. ### Compact supports Deﬁnition A homology theory has compact supports on

a class of topological spaces C if, for every natural number p and every space X in C, Hp(X) is isomorphic to the direct limit of Hp(K) for compact subspaces K of X via the map induced by the inclusion maps K → X. Mardeˇ si´ c and Prasolov also showed that if strong homology has countable supports on closed subsets of Euclidean space, then limn A = 0 for all n ≥ 1.
10. ### The inverse system A Given a function f ∈ ωω,

let I(f ) = {(k, m) ∈ ω × ω | m ≤ f (k)} , and let Af = I(f ) Z.
11. ### The inverse system A Given a function f ∈ ωω,

let I(f ) = {(k, m) ∈ ω × ω | m ≤ f (k)} , and let Af = I(f ) Z. If f ≤ g, we have a projection map pfg : Ag → Af .
12. ### The inverse system A Given a function f ∈ ωω,

let I(f ) = {(k, m) ∈ ω × ω | m ≤ f (k)} , and let Af = I(f ) Z. If f ≤ g, we have a projection map pfg : Ag → Af . A is the inverse system Af , pfg f , g ∈ ωω, f ≤ g .
13. ### The inverse system A Given a function f ∈ ωω,

let I(f ) = {(k, m) ∈ ω × ω | m ≤ f (k)} , and let Af = I(f ) Z. If f ≤ g, we have a projection map pfg : Ag → Af . A is the inverse system Af , pfg f , g ∈ ωω, f ≤ g . Its inverse limit, lim0 A, is ω ω Z.
14. ### Condensed mathematics The question of the consistency of limn A

= 0 for n ≥ 1 arose independently in recent work of Clausen and Scholze on condensed mathematics, a new approach to doing algebra in situations in which the algebraic structures carry topologies.
15. ### Condensed mathematics The question of the consistency of limn A

= 0 for n ≥ 1 arose independently in recent work of Clausen and Scholze on condensed mathematics, a new approach to doing algebra in situations in which the algebraic structures carry topologies. They introduced the category of condensed abelian groups as a replacement for topological abelian groups.
16. ### Condensed mathematics The question of the consistency of limn A

= 0 for n ≥ 1 arose independently in recent work of Clausen and Scholze on condensed mathematics, a new approach to doing algebra in situations in which the algebraic structures carry topologies. They introduced the category of condensed abelian groups as a replacement for topological abelian groups. The natural question of whether pro-abelian groups embed fully faithfully into condensed abelian groups is equivalent, in its simplest case, to the question of whether limn A = 0 for n ≥ 1.

18. ### Nontrivial coherent families Given f , g ∈ ωω, let

f ∧ g denote their greatest lower bound in (ωω, ≤).
19. ### Nontrivial coherent families Given f , g ∈ ωω, let

f ∧ g denote their greatest lower bound in (ωω, ≤). Deﬁnition Let Φ = {ϕf : I(f ) → Z | f ∈ ωω} be a family of functions.
20. ### Nontrivial coherent families Given f , g ∈ ωω, let

f ∧ g denote their greatest lower bound in (ωω, ≤). Deﬁnition Let Φ = {ϕf : I(f ) → Z | f ∈ ωω} be a family of functions. 1 Φ is coherent if ϕf I(f ∧ g) =∗ ϕg I(f ∧ g) for all f , g ∈ ωω.
21. ### Nontrivial coherent families Given f , g ∈ ωω, let

f ∧ g denote their greatest lower bound in (ωω, ≤). Deﬁnition Let Φ = {ϕf : I(f ) → Z | f ∈ ωω} be a family of functions. 1 Φ is coherent if ϕf I(f ∧ g) =∗ ϕg I(f ∧ g) for all f , g ∈ ωω. 2 Φ is trivial if there is a function ψ : ω × ω → Z such that ϕf =∗ ψ I(f ) for all f ∈ ωω.
22. ### Nontrivial coherent families Given f , g ∈ ωω, let

f ∧ g denote their greatest lower bound in (ωω, ≤). Deﬁnition Let Φ = {ϕf : I(f ) → Z | f ∈ ωω} be a family of functions. 1 Φ is coherent if ϕf I(f ∧ g) =∗ ϕg I(f ∧ g) for all f , g ∈ ωω. 2 Φ is trivial if there is a function ψ : ω × ω → Z such that ϕf =∗ ψ I(f ) for all f ∈ ωω. Clearly, a trivial family is coherent.
23. ### Nontrivial coherent families Given f , g ∈ ωω, let

f ∧ g denote their greatest lower bound in (ωω, ≤). Deﬁnition Let Φ = {ϕf : I(f ) → Z | f ∈ ωω} be a family of functions. 1 Φ is coherent if ϕf I(f ∧ g) =∗ ϕg I(f ∧ g) for all f , g ∈ ωω. 2 Φ is trivial if there is a function ψ : ω × ω → Z such that ϕf =∗ ψ I(f ) for all f ∈ ωω. Clearly, a trivial family is coherent. A nontrivial coherent family Φ is a clear example of set theoretic incompactness: each local family {ϕf | f < g} (for a ﬁxed g ∈ ωω) is trivial, as witnessed by ϕg itself, but the entire family is not.
24. ### 2-dimensional nontrivial coherence Deﬁnition Let Φ = ϕfg : I(f

∧ g) → Z f , g ∈ ωω .
25. ### 2-dimensional nontrivial coherence Deﬁnition Let Φ = ϕfg : I(f

∧ g) → Z f , g ∈ ωω . 1 Φ is alternating if ϕfg = −ϕgf for all f , g ∈ ωω.
26. ### 2-dimensional nontrivial coherence Deﬁnition Let Φ = ϕfg : I(f

∧ g) → Z f , g ∈ ωω . 1 Φ is alternating if ϕfg = −ϕgf for all f , g ∈ ωω. 2 Φ is 2-coherent if it is alternating and ϕfg + ϕgh =∗ ϕfh for all f , g, h ∈ ωω. (All functions restricted to I(f ∧ g ∧ h).)
27. ### 2-dimensional nontrivial coherence Deﬁnition Let Φ = ϕfg : I(f

∧ g) → Z f , g ∈ ωω . 1 Φ is alternating if ϕfg = −ϕgf for all f , g ∈ ωω. 2 Φ is 2-coherent if it is alternating and ϕfg + ϕgh =∗ ϕfh for all f , g, h ∈ ωω. (All functions restricted to I(f ∧ g ∧ h).) 3 Φ is 2-trivial if there is a family Ψ = {ψf : I(f ) → Z | f ∈ ωω} such that ψg − ψf =∗ ϕfg for all f , g ∈ ωω.
28. ### 2-dimensional nontrivial coherence Deﬁnition Let Φ = ϕfg : I(f

∧ g) → Z f , g ∈ ωω . 1 Φ is alternating if ϕfg = −ϕgf for all f , g ∈ ωω. 2 Φ is 2-coherent if it is alternating and ϕfg + ϕgh =∗ ϕfh for all f , g, h ∈ ωω. (All functions restricted to I(f ∧ g ∧ h).) 3 Φ is 2-trivial if there is a family Ψ = {ψf : I(f ) → Z | f ∈ ωω} such that ψg − ψf =∗ ϕfg for all f , g ∈ ωω. Again, a 2-trivial family is 2-coherent.
29. ### 2-dimensional nontrivial coherence Deﬁnition Let Φ = ϕfg : I(f

∧ g) → Z f , g ∈ ωω . 1 Φ is alternating if ϕfg = −ϕgf for all f , g ∈ ωω. 2 Φ is 2-coherent if it is alternating and ϕfg + ϕgh =∗ ϕfh for all f , g, h ∈ ωω. (All functions restricted to I(f ∧ g ∧ h).) 3 Φ is 2-trivial if there is a family Ψ = {ψf : I(f ) → Z | f ∈ ωω} such that ψg − ψf =∗ ϕfg for all f , g ∈ ωω. Again, a 2-trivial family is 2-coherent. A non-2-trivial 2-coherent family Φ is an example of incompactness: each local family {ϕfg | f , g < h} (for a ﬁxed h ∈ ωω) is 2-trivial, as witnessed by the family {−ϕfh | f < h}, but the entire family is not.
30. ### n-dimensional nontrivial coherence Given a sequence f = (f0, .

. . , fn−1) and i < n, f i is the sequence of length n − 1 formed by removing fi from f . Deﬁnition Fix n ≥ 2, and let Φ = ϕ f : I(∧f ) → Z f ∈ (ωω)n .
31. ### n-dimensional nontrivial coherence Given a sequence f = (f0, .

. . , fn−1) and i < n, f i is the sequence of length n − 1 formed by removing fi from f . Deﬁnition Fix n ≥ 2, and let Φ = ϕ f : I(∧f ) → Z f ∈ (ωω)n . 1 Φ is alternating if ϕ f = sgn(σ)ϕ σ(f ) for all f ∈ (ωω)n and all permutations σ.
32. ### n-dimensional nontrivial coherence Given a sequence f = (f0, .

. . , fn−1) and i < n, f i is the sequence of length n − 1 formed by removing fi from f . Deﬁnition Fix n ≥ 2, and let Φ = ϕ f : I(∧f ) → Z f ∈ (ωω)n . 1 Φ is alternating if ϕ f = sgn(σ)ϕ σ(f ) for all f ∈ (ωω)n and all permutations σ. 2 Φ is n-coherent if it is alternating and n i=0 (−1)i ϕ f i =∗ 0 for all f ∈ (ωω)n+1.
33. ### n-dimensional nontrivial coherence Given a sequence f = (f0, .

. . , fn−1) and i < n, f i is the sequence of length n − 1 formed by removing fi from f . Deﬁnition Fix n ≥ 2, and let Φ = ϕ f : I(∧f ) → Z f ∈ (ωω)n . 1 Φ is alternating if ϕ f = sgn(σ)ϕ σ(f ) for all f ∈ (ωω)n and all permutations σ. 2 Φ is n-coherent if it is alternating and n i=0 (−1)i ϕ f i =∗ 0 for all f ∈ (ωω)n+1. 3 Φ is n-trivial if there is an alternating family ψ f : I(∧f ) → Z f ∈ (ωω)n−1 such that n−1 i=0 (−1)i ψ f i =∗ ϕ f for all f ∈ (ωω)n.
34. ### limn A and nontrivial coherence Coherent and trivial can now

be thought of as 1-coherent and 1-trivial.
35. ### limn A and nontrivial coherence Coherent and trivial can now

be thought of as 1-coherent and 1-trivial. Theorem (Mardeˇ si´ c-Prasolov (n = 1), Bergfalk (n ≥ 2)) Fix n ≥ 1. Then limn A = 0 if and only if every n-coherent family Φ = ϕ f f ∈ (ωω)n is n-trivial.
36. ### limn A and nontrivial coherence Coherent and trivial can now

be thought of as 1-coherent and 1-trivial. Theorem (Mardeˇ si´ c-Prasolov (n = 1), Bergfalk (n ≥ 2)) Fix n ≥ 1. Then limn A = 0 if and only if every n-coherent family Φ = ϕ f f ∈ (ωω)n is n-trivial. Thus, to prove that limn A = 0, it suﬃces to show that every n-coherent family is n-trivial.

lim1 A = 0.
38. ### lim1 A Theorem (Mardeˇ si´ c-Prasolov, Simon, ’88) CH ⇒

lim1 A = 0. Theorem (Dow-Simon-Vaughan, ‘89) d = ℵ1 ⇒ lim1 A = 0.
39. ### lim1 A Theorem (Mardeˇ si´ c-Prasolov, Simon, ’88) CH ⇒

lim1 A = 0. Theorem (Dow-Simon-Vaughan, ‘89) d = ℵ1 ⇒ lim1 A = 0. PFA ⇒ lim1 A = 0.
40. ### lim1 A Theorem (Mardeˇ si´ c-Prasolov, Simon, ’88) CH ⇒

lim1 A = 0. Theorem (Dow-Simon-Vaughan, ‘89) d = ℵ1 ⇒ lim1 A = 0. PFA ⇒ lim1 A = 0. Theorem (Todorcevic, ‘98) OCA ⇒ lim1 A = 0.
41. ### lim1 A Theorem (Mardeˇ si´ c-Prasolov, Simon, ’88) CH ⇒

lim1 A = 0. Theorem (Dow-Simon-Vaughan, ‘89) d = ℵ1 ⇒ lim1 A = 0. PFA ⇒ lim1 A = 0. Theorem (Todorcevic, ‘98) OCA ⇒ lim1 A = 0. MAℵ1 ⇒ lim1 A = 0.
42. ### lim1 A Theorem (Mardeˇ si´ c-Prasolov, Simon, ’88) CH ⇒

lim1 A = 0. Theorem (Dow-Simon-Vaughan, ‘89) d = ℵ1 ⇒ lim1 A = 0. PFA ⇒ lim1 A = 0. Theorem (Todorcevic, ‘98) OCA ⇒ lim1 A = 0. MAℵ1 ⇒ lim1 A = 0. Theorem (Kamo, ‘93) V Add(ω,ω2) |= “ lim1 A = 0”.

0.
44. ### Higher limits Theorem (Bergfalk, ‘17) PFA ⇒ lim2 A =

0. Theorem (Bergfalk-LH, ‘19) Suppose that κ is a measurable cardinal and that P is a length-κ ﬁnite support iteration of Hechler forcings. Then V P |= “ lim nA = 0 for all n ≥ 1”.

46. ### Hechler forcing Recall that conditions of Hechler forcing are of

the form p = (sp, f p), where sp ∈ <ωω and f p ∈ ωω.
47. ### Hechler forcing Recall that conditions of Hechler forcing are of

the form p = (sp, f p), where sp ∈ <ωω and f p ∈ ωω. If p and q are Hechler conditions, then q ≤ p if and only if • sq end-extends sp; • f q ≥ f p; • sq(k) > f p(k) for all k ∈ dom(sq) \ dom(qp).
48. ### Hechler forcing Recall that conditions of Hechler forcing are of

the form p = (sp, f p), where sp ∈ <ωω and f p ∈ ωω. If p and q are Hechler conditions, then q ≤ p if and only if • sq end-extends sp; • f q ≥ f p; • sq(k) > f p(k) for all k ∈ dom(sq) \ dom(qp). In what follows, P denotes a length-κ ﬁnite support iteration of Hechler forcings, where κ is a ﬁxed regular uncountable cardinal.
49. ### Hechler forcing Recall that conditions of Hechler forcing are of

the form p = (sp, f p), where sp ∈ <ωω and f p ∈ ωω. If p and q are Hechler conditions, then q ≤ p if and only if • sq end-extends sp; • f q ≥ f p; • sq(k) > f p(k) for all k ∈ dom(sq) \ dom(qp). In what follows, P denotes a length-κ ﬁnite support iteration of Hechler forcings, where κ is a ﬁxed regular uncountable cardinal. As a result of a simple ∆-system argument, we know that, if pα | α < κ is a sequence of conditions in P, then there is an unbounded set A ⊆ κ such that pα | α ∈ A consists of pairwise compatible conditions.
50. ### A ﬁrst attempt In our proof, we will be working

with families of conditions indexed by ﬁnite increasing sequences from κ. We would like to use a multi-dimensional ∆-system argument to perform a similar sort of uniformization. A na¨ ıve ﬁrst attempt might be:
51. ### A ﬁrst attempt In our proof, we will be working

with families of conditions indexed by ﬁnite increasing sequences from κ. We would like to use a multi-dimensional ∆-system argument to perform a similar sort of uniformization. A na¨ ıve ﬁrst attempt might be: Lemma? Suppose that κ is suﬃciently large (e.g., measurable) and ﬁx n ≥ 1. Given any family pα | α ∈ [κ]n from P, there is an unbounded A ⊆ κ such that pα | α ∈ [A]n consists of pairwise compatible conditions.
52. ### A ﬁrst attempt In our proof, we will be working

with families of conditions indexed by ﬁnite increasing sequences from κ. We would like to use a multi-dimensional ∆-system argument to perform a similar sort of uniformization. A na¨ ıve ﬁrst attempt might be: Lemma? Suppose that κ is suﬃciently large (e.g., measurable) and ﬁx n ≥ 1. Given any family pα | α ∈ [κ]n from P, there is an unbounded A ⊆ κ such that pα | α ∈ [A]n consists of pairwise compatible conditions. This is obviously false. If n = 2, for example, we can deﬁne pαβ | α < β < κ such that pαβ(α) is (a name for) ( 0 , 0) and pαβ(β) is ( 1 , 0). Then, for any α < β < γ, we have pαβ ⊥ pβγ.
53. ### A ﬁrst attempt In our proof, we will be working

with families of conditions indexed by ﬁnite increasing sequences from κ. We would like to use a multi-dimensional ∆-system argument to perform a similar sort of uniformization. A na¨ ıve ﬁrst attempt might be: Lemma? Suppose that κ is suﬃciently large (e.g., measurable) and ﬁx n ≥ 1. Given any family pα | α ∈ [κ]n from P, there is an unbounded A ⊆ κ such that pα | α ∈ [A]n consists of pairwise compatible conditions. This is obviously false. If n = 2, for example, we can deﬁne pαβ | α < β < κ such that pαβ(α) is (a name for) ( 0 , 0) and pαβ(β) is ( 1 , 0). Then, for any α < β < γ, we have pαβ ⊥ pβγ. This turns out to be the only obstacle, though.
54. ### Aligned sets Deﬁnition Fix n ≥ 1 and suppose that

α = α0, . . . , αn−1 and β = β0, . . . , βn−1 are in [κ]n (and hence are increasing). We say that α and β are aligned if, for all i, j < n, αi = βj ⇒ i = j.
55. ### Aligned sets Deﬁnition Fix n ≥ 1 and suppose that

α = α0, . . . , αn−1 and β = β0, . . . , βn−1 are in [κ]n (and hence are increasing). We say that α and β are aligned if, for all i, j < n, αi = βj ⇒ i = j. Lemma Suppose that κ is weakly compact, and ﬁx n ≥ 1. Given any family pα | α ∈ [κ]n , there is an unbounded A ⊆ κ such that, for all α, β ∈ [κ]n, if α and β are aligned, then pα and p β are compatible.
56. ### Aligned sets Deﬁnition Fix n ≥ 1 and suppose that

α = α0, . . . , αn−1 and β = β0, . . . , βn−1 are in [κ]n (and hence are increasing). We say that α and β are aligned if, for all i, j < n, αi = βj ⇒ i = j. Lemma Suppose that κ is weakly compact, and ﬁx n ≥ 1. Given any family pα | α ∈ [κ]n , there is an unbounded A ⊆ κ such that, for all α, β ∈ [κ]n, if α and β are aligned, then pα and p β are compatible. We actually require a bit more than this, for which we seem to need a measurable cardinal.
57. ### The ∆-system lemma Lemma Suppose that κ is measurable, with

normal measure U, and ﬁx n ≥ 1. Let P be a length-κ ﬁnite support iteration of Hechler forcings, and let {pα | α ∈ [κ]n} be a family of conditions in P.
58. ### The ∆-system lemma Lemma Suppose that κ is measurable, with

normal measure U, and ﬁx n ≥ 1. Let P be a length-κ ﬁnite support iteration of Hechler forcings, and let {pα | α ∈ [κ]n} be a family of conditions in P. Then there is a set A ∈ U and conditions {pα | α ∈ [A]<n} such that
59. ### The ∆-system lemma Lemma Suppose that κ is measurable, with

normal measure U, and ﬁx n ≥ 1. Let P be a length-κ ﬁnite support iteration of Hechler forcings, and let {pα | α ∈ [κ]n} be a family of conditions in P. Then there is a set A ∈ U and conditions {pα | α ∈ [A]<n} such that • for all α β in [A]≤n, we have dom(pα ) dom(p β ) and pα = p β dom(pα );
60. ### The ∆-system lemma Lemma Suppose that κ is measurable, with

normal measure U, and ﬁx n ≥ 1. Let P be a length-κ ﬁnite support iteration of Hechler forcings, and let {pα | α ∈ [κ]n} be a family of conditions in P. Then there is a set A ∈ U and conditions {pα | α ∈ [A]<n} such that • for all α β in [A]≤n, we have dom(pα ) dom(p β ) and pα = p β dom(pα ); • for all α ∈ [A]<n, the set dom(pα β ) β ∈ A \ max(α + 1) forms a ∆-system with root dom(pα );
61. ### The ∆-system lemma Lemma Suppose that κ is measurable, with

normal measure U, and ﬁx n ≥ 1. Let P be a length-κ ﬁnite support iteration of Hechler forcings, and let {pα | α ∈ [κ]n} be a family of conditions in P. Then there is a set A ∈ U and conditions {pα | α ∈ [A]<n} such that • for all α β in [A]≤n, we have dom(pα ) dom(p β ) and pα = p β dom(pα ); • for all α ∈ [A]<n, the set dom(pα β ) β ∈ A \ max(α + 1) forms a ∆-system with root dom(pα ); • for all α, β ∈ [A]≤n of equal length, if α and β are aligned, then dom(pα ) and dom(p β ) are aligned.

63. ### Proof sketch (n = 2) Main Theorem Suppose that κ

is a measurable cardinal and that P is a length-κ ﬁnite support iteration of Hechler forcings. In V P, for every n ≥ 1, every n-coherent family ϕ f f ∈ (ωω)n is n-trivial.
64. ### Proof sketch (n = 2) Main Theorem Suppose that κ

is a measurable cardinal and that P is a length-κ ﬁnite support iteration of Hechler forcings. In V P, for every n ≥ 1, every n-coherent family ϕ f f ∈ (ωω)n is n-trivial. Proof sketch for n = 2: We start by making some simplifying assumptions.
65. ### Proof sketch (n = 2) Main Theorem Suppose that κ

is a measurable cardinal and that P is a length-κ ﬁnite support iteration of Hechler forcings. In V P, for every n ≥ 1, every n-coherent family ϕ f f ∈ (ωω)n is n-trivial. Proof sketch for n = 2: We start by making some simplifying assumptions. • To show that a 2-coherent family of functions is trivial, it suﬃces to ﬁnd a trivialization indexed by some ≤∗-coﬁnal family of functions.
66. ### Proof sketch (n = 2) Main Theorem Suppose that κ

is a measurable cardinal and that P is a length-κ ﬁnite support iteration of Hechler forcings. In V P, for every n ≥ 1, every n-coherent family ϕ f f ∈ (ωω)n is n-trivial. Proof sketch for n = 2: We start by making some simplifying assumptions. • To show that a 2-coherent family of functions is trivial, it suﬃces to ﬁnd a trivialization indexed by some ≤∗-coﬁnal family of functions. • Since the Hechler reals fα | α < κ are ≤∗-increasing and coﬁnal, it suﬃces to ﬁnd a trivialization indexed by fα | α ∈ B for some coﬁnal B ⊆ κ.
67. ### • The 2-triviality of a 2-coherent family ϕfg f ,

g ∈ ωω is equivalent to the following statement:
68. ### • The 2-triviality of a 2-coherent family ϕfg f ,

g ∈ ωω is equivalent to the following statement: There is an alternating family ψfg f , g ∈ ωω such that each ψfg : I(f ∧ g) → Z is ﬁnitely supported and, for all f , g, h ∈ ωω, (ϕgh − ψgh) − (ϕfh − ψfh) + (ϕfg − ψfg) = 0.
69. ### • The 2-triviality of a 2-coherent family ϕfg f ,

g ∈ ωω is equivalent to the following statement: There is an alternating family ψfg f , g ∈ ωω such that each ψfg : I(f ∧ g) → Z is ﬁnitely supported and, for all f , g, h ∈ ωω, (ϕgh − ψgh) − (ϕfh − ψfh) + (ϕfg − ψfg) = 0. In V , ﬁx a condition p ∈ P and a P-name ˙ Φ = ˙ ϕαβ α, β < κ for a 2-coherent family indexed by pairs of Hechler reals.
70. ### • The 2-triviality of a 2-coherent family ϕfg f ,

g ∈ ωω is equivalent to the following statement: There is an alternating family ψfg f , g ∈ ωω such that each ψfg : I(f ∧ g) → Z is ﬁnitely supported and, for all f , g, h ∈ ωω, (ϕgh − ψgh) − (ϕfh − ψfh) + (ϕfg − ψfg) = 0. In V , ﬁx a condition p ∈ P and a P-name ˙ Φ = ˙ ϕαβ α, β < κ for a 2-coherent family indexed by pairs of Hechler reals. For α < β < γ < κ above the domain of p, let ˙ e(α, β, γ) be a P name for ϕβγ − ϕαγ + ϕαβ, let ˙ ¯ e(α, β, γ) be a P-name for its restriction to its (ﬁnite) support, and ﬁx pαβγ ≤ p such that
71. ### • The 2-triviality of a 2-coherent family ϕfg f ,

g ∈ ωω is equivalent to the following statement: There is an alternating family ψfg f , g ∈ ωω such that each ψfg : I(f ∧ g) → Z is ﬁnitely supported and, for all f , g, h ∈ ωω, (ϕgh − ψgh) − (ϕfh − ψfh) + (ϕfg − ψfg) = 0. In V , ﬁx a condition p ∈ P and a P-name ˙ Φ = ˙ ϕαβ α, β < κ for a 2-coherent family indexed by pairs of Hechler reals. For α < β < γ < κ above the domain of p, let ˙ e(α, β, γ) be a P name for ϕβγ − ϕαγ + ϕαβ, let ˙ ¯ e(α, β, γ) be a P-name for its restriction to its (ﬁnite) support, and ﬁx pαβγ ≤ p such that • pαβγ fα ≤ fβ ≤ fγ;
72. ### • The 2-triviality of a 2-coherent family ϕfg f ,

g ∈ ωω is equivalent to the following statement: There is an alternating family ψfg f , g ∈ ωω such that each ψfg : I(f ∧ g) → Z is ﬁnitely supported and, for all f , g, h ∈ ωω, (ϕgh − ψgh) − (ϕfh − ψfh) + (ϕfg − ψfg) = 0. In V , ﬁx a condition p ∈ P and a P-name ˙ Φ = ˙ ϕαβ α, β < κ for a 2-coherent family indexed by pairs of Hechler reals. For α < β < γ < κ above the domain of p, let ˙ e(α, β, γ) be a P name for ϕβγ − ϕαγ + ϕαβ, let ˙ ¯ e(α, β, γ) be a P-name for its restriction to its (ﬁnite) support, and ﬁx pαβγ ≤ p such that • pαβγ fα ≤ fβ ≤ fγ; • pαβγ decides the value of ˙ ¯ e(α, β, γ).
73. ### Apply our ∆-system lemma to obtain A ∈ U and

conditions {pα | α ∈ [A]<3} such that
74. ### Apply our ∆-system lemma to obtain A ∈ U and

conditions {pα | α ∈ [A]<3} such that • for all α β in [A]≤3, we have dom(pα ) dom(p β ) and pα = p β dom(pα );
75. ### Apply our ∆-system lemma to obtain A ∈ U and

conditions {pα | α ∈ [A]<3} such that • for all α β in [A]≤3, we have dom(pα ) dom(p β ) and pα = p β dom(pα ); • for all α ∈ [A]<n, the set dom(pα β ) β ∈ A \ max(α + 1) forms a ∆-system with root dom(pα );
76. ### Apply our ∆-system lemma to obtain A ∈ U and

conditions {pα | α ∈ [A]<3} such that • for all α β in [A]≤3, we have dom(pα ) dom(p β ) and pα = p β dom(pα ); • for all α ∈ [A]<n, the set dom(pα β ) β ∈ A \ max(α + 1) forms a ∆-system with root dom(pα ); • for all α, β ∈ [A]≤n of equal length, if α and β are aligned, then dom(pα ) and dom(p β ) are aligned.
77. ### Apply our ∆-system lemma to obtain A ∈ U and

conditions {pα | α ∈ [A]<3} such that • for all α β in [A]≤3, we have dom(pα ) dom(p β ) and pα = p β dom(pα ); • for all α ∈ [A]<n, the set dom(pα β ) β ∈ A \ max(α + 1) forms a ∆-system with root dom(pα ); • for all α, β ∈ [A]≤n of equal length, if α and β are aligned, then dom(pα ) and dom(p β ) are aligned. Using the measurability of κ, we can also assume that there is a ﬁxed ¯ e such that pαβγ ˙ ¯ e(α, β, γ) = ¯ e for all α < β < γ in A.

79. ### We claim that p∅ forces ˙ Φ to be trivial.

In V , partition A into unbounded sets Γ0, Γ1, Γ2.
80. ### We claim that p∅ forces ˙ Φ to be trivial.

In V , partition A into unbounded sets Γ0, Γ1, Γ2. Let G be P-generic with p∅ ∈ G. The following facts, which hold in V [G], are the point of our ∆-system lemma:
81. ### We claim that p∅ forces ˙ Φ to be trivial.

In V , partition A into unbounded sets Γ0, Γ1, Γ2. Let G be P-generic with p∅ ∈ G. The following facts, which hold in V [G], are the point of our ∆-system lemma: • B := {α ∈ Γ0 | pα ∈ G} is unbounded.
82. ### We claim that p∅ forces ˙ Φ to be trivial.

In V , partition A into unbounded sets Γ0, Γ1, Γ2. Let G be P-generic with p∅ ∈ G. The following facts, which hold in V [G], are the point of our ∆-system lemma: • B := {α ∈ Γ0 | pα ∈ G} is unbounded. • For all α < β in B, there is αβ ∈ Γ1 \ β such that pα αβ , pβ αβ ∈ G.
83. ### We claim that p∅ forces ˙ Φ to be trivial.

In V , partition A into unbounded sets Γ0, Γ1, Γ2. Let G be P-generic with p∅ ∈ G. The following facts, which hold in V [G], are the point of our ∆-system lemma: • B := {α ∈ Γ0 | pα ∈ G} is unbounded. • For all α < β in B, there is αβ ∈ Γ1 \ β such that pα αβ , pβ αβ ∈ G. • For all α < β < γ in B, there is αβγ ∈ Γ2 \ max{ αβ, αγ, βγ} such that pα αβ αβγ , pα αγ αβγ , pβ αβ αβγ , pβ βγ αβγ , pγ αγ, αβγ , pγ βγ, αβγ ∈ G.
84. ### For α < β in B, let ψαβ := e(α,

β, αβ) = ϕβ αβ − ϕα αβ + ϕαβ.
85. ### For α < β in B, let ψαβ := e(α,

β, αβ) = ϕβ αβ − ϕα αβ + ϕαβ. We claim that this works. To see this, ﬁx α < β < γ in B. We must show that e(α, β, γ) = ψβγ − ψαγ + ψαβ.
86. ### For α < β in B, let ψαβ := e(α,

β, αβ) = ϕβ αβ − ϕα αβ + ϕαβ. We claim that this works. To see this, ﬁx α < β < γ in B. We must show that e(α, β, γ) = ψβγ − ψαγ + ψαβ. NOTE: For any α < β < γ < δ < κ, we have e(β, γ, δ) − e(α, γ, δ) + e(α, β, δ) − e(α, β, γ) = 0.
87. ### For α < β in B, let ψαβ := e(α,

β, αβ) = ϕβ αβ − ϕα αβ + ϕαβ. We claim that this works. To see this, ﬁx α < β < γ in B. We must show that e(α, β, γ) = ψβγ − ψαγ + ψαβ. NOTE: For any α < β < γ < δ < κ, we have e(β, γ, δ) − e(α, γ, δ) + e(α, β, δ) − e(α, β, γ) = 0. Thus, e(α, β, γ) = e(β, γ, αβγ) − e(α, γ, αβγ) + e(α, β, αβγ).
88. ### For α < β in B, let ψαβ := e(α,

β, αβ) = ϕβ αβ − ϕα αβ + ϕαβ. We claim that this works. To see this, ﬁx α < β < γ in B. We must show that e(α, β, γ) = ψβγ − ψαγ + ψαβ. NOTE: For any α < β < γ < δ < κ, we have e(β, γ, δ) − e(α, γ, δ) + e(α, β, δ) − e(α, β, γ) = 0. Thus, e(α, β, γ) = e(β, γ, αβγ) − e(α, γ, αβγ) + e(α, β, αβγ). e(β, γ, αβγ) = −e(γ, βγ, αβγ) + e(β, βγ, αβγ) + e(β, γ, βγ) = −e + e + ψβγ = ψβγ
89. ### For α < β in B, let ψαβ := e(α,

β, αβ) = ϕβ αβ − ϕα αβ + ϕαβ. We claim that this works. To see this, ﬁx α < β < γ in B. We must show that e(α, β, γ) = ψβγ − ψαγ + ψαβ. NOTE: For any α < β < γ < δ < κ, we have e(β, γ, δ) − e(α, γ, δ) + e(α, β, δ) − e(α, β, γ) = 0. Thus, e(α, β, γ) = e(β, γ, αβγ) − e(α, γ, αβγ) + e(α, β, αβγ). e(β, γ, αβγ) = −e(γ, βγ, αβγ) + e(β, βγ, αβγ) + e(β, γ, βγ) = −e + e + ψβγ = ψβγ Similarly for e(α, γ, αβγ) and e(α, β, αβγ).
90. ### For α < β in B, let ψαβ := e(α,

β, αβ) = ϕβ αβ − ϕα αβ + ϕαβ. We claim that this works. To see this, ﬁx α < β < γ in B. We must show that e(α, β, γ) = ψβγ − ψαγ + ψαβ. NOTE: For any α < β < γ < δ < κ, we have e(β, γ, δ) − e(α, γ, δ) + e(α, β, δ) − e(α, β, γ) = 0. Thus, e(α, β, γ) = e(β, γ, αβγ) − e(α, γ, αβγ) + e(α, β, αβγ). e(β, γ, αβγ) = −e(γ, βγ, αβγ) + e(β, βγ, αβγ) + e(β, γ, βγ) = −e + e + ψβγ = ψβγ Similarly for e(α, γ, αβγ) and e(α, β, αβγ). This reduces to e(α, β, γ) = ψβγ − ψαγ + ψαβ, as desired.

92. ### Optimality There are a number of questions one can ask

about the optimality of our theorem.
93. ### Optimality There are a number of questions one can ask

about the optimality of our theorem. Question What is the consistency strength of the statement “ limn A = 0 for all n ≥ 1”?
94. ### Optimality There are a number of questions one can ask

about the optimality of our theorem. Question What is the consistency strength of the statement “ limn A = 0 for all n ≥ 1”? Question How small can the continuum be in a model of “ limn A = 0 for all n ≥ 1”?
95. ### Optimality There are a number of questions one can ask

about the optimality of our theorem. Question What is the consistency strength of the statement “ limn A = 0 for all n ≥ 1”? Question How small can the continuum be in a model of “ limn A = 0 for all n ≥ 1”? We conjecture that it must be greater than ℵω. A natural place to look is the model obtained from adding ℵω+1 Cohen reals to a model of CH.
96. ### Strong homology Question Is it consistent that strong homology is

additive on some nice class of topological spaces, e.g., locally compact metric spaces?
97. ### Strong homology Question Is it consistent that strong homology is

additive on some nice class of topological spaces, e.g., locally compact metric spaces? Question Is it consistent that strong homology has compact supports on some nice class of topological spaces, e.g., locally compact metric spaces?
98. ### Strong homology Question Is it consistent that strong homology is

additive on some nice class of topological spaces, e.g., locally compact metric spaces? Question Is it consistent that strong homology has compact supports on some nice class of topological spaces, e.g., locally compact metric spaces? The model for our Main Theorem is a natural ﬁrst place to look for these questions.
99. ### Results contained in “Simultaneously vanishing higher derived limits,” joint with

Jeﬀrey Bergfalk. Available at: https://arxiv.org/abs/1907.11744 All artwork by Josef Albers, inspired by visits to Monte Alban