Simultaneously vanishing higher derived limits

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

I. Homological beginnings

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 .

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.

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.

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.

A Hawaiian earring

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.

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.

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.

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 .

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 .

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.

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.

= 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.

= 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.

II. Set theoretic reformulations

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

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

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.

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.

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.

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.

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

∧ g) → Z f , g ∈ ωω . 1 Φ is alternating if ϕfg = −ϕgf for all f , g ∈ ωω.

∧ 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).)

∧ 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.

∧ 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.

∧ 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.

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 .

. . , 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.

. . , 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.

. . , 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.

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

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.

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

lim1 A = 0. Theorem (Dow-Simon-Vaughan, ‘89) d = ℵ1 ⇒ lim1 A = 0.

lim1 A = 0. Theorem (Dow-Simon-Vaughan, ‘89) d = ℵ1 ⇒ lim1 A = 0. PFA ⇒ lim1 A = 0.

lim1 A = 0. Theorem (Dow-Simon-Vaughan, ‘89) d = ℵ1 ⇒ lim1 A = 0. PFA ⇒ lim1 A = 0. Theorem (Todorcevic, ‘98) OCA ⇒ lim1 A = 0.

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.

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”.

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

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”.

III. A strong ∆-system lemma

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

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).

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.

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.

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:

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.

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.

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.

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.

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.

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.

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.

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

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 );

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 );

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.

IV. Vanishing higher limits

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.

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.

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.

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.

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

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.

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.

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

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;

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).

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

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 );

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 );

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.

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.

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

In V , partition A into unbounded sets Γ0, Γ1, Γ2.

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:

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.

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.

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.

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

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.

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.

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).

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

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).

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.

V. Further directions

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

about the optimality of our theorem. Question What is the consistency strength of the statement “ limn A = 0 for all n ≥ 1”?

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”?

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.

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

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?

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.

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

Thank you!

Simultaneously vanishing higher derived limits

Simultaneously vanishing higher derived limits

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