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One of the better-known independence results in general mathematics is Shelah's solution to Whitehead's problem of whether $\mathrm{Ext}^1(A,\mathbb{Z})=0$ implies that an abelian group $A$ is free. The point of departure for the present work is Clausen and Scholze's proof that, in contrast, one natural interpretation of Whitehead's problem within their recently-developed framework of condensed mathematics has an affirmative answer in $\mathsf{ZFC}$. We record two alternative proofs of this result, as well as several original variations on it, both for their intrinsic interest and as a springboard for a broader study of the relations between condensed mathematics and set theoretic forcing. We show more particularly how the condensation $\underline{X}$ of any locally compact Hausdorff space $X$ may be viewed as an organized presentation of the forcing names for the points of canonical interpretations of $X$ in all possible set-forcing extensions of the universe, and we argue our main result by way of this fact. We show also that when interpreted within the category of light condensed abelian groups, Whitehead's problem is again independent of the $\mathsf{ZFC}$ axioms. In fact we show that it is consistent that Whitehead's problem has a negative solution within the category of $\kappa$-condensed abelian groups for every uncountable cardinal $\kappa$, but that this scenario, in turn, is inconsistent with the existence of a strongly compact cardinal., Comment: 40 pages, corrected statement of Theorem 6.3

This is the second installment in a series of papers applying descriptive set theoretic techniques to both analyze and enrich classical functors from homological algebra and algebraic topology. In it, we show that the \v{C}ech cohomology functors $\check{\mathrm{H}}^n$ on the category of locally compact separable metric spaces each factor into (i) what we term their definable version, a functor $\check{\mathrm{H}}^n_{\mathrm{def}}$ taking values in the category $\mathsf{GPC}$ of groups with a Polish cover (a category first introduced in this work's predecessor), followed by (ii) a forgetful functor from $\mathsf{GPC}$ to the category of groups. These definable cohomology functors powerfully refine their classical counterparts: we show that they are complete invariants, for example, of the homotopy types of mapping telescopes of $d$-spheres or $d$-tori for any $d\geq 1$, and, in contrast, that there exist uncountable families of pairwise homotopy inequivalent mapping telescopes of either sort on which the classical cohomology functors are constant. We then apply the functors $\check{\mathrm{H}}^n_{\mathrm{def}}$ to show that a seminal problem in the development of algebraic topology, namely Borsuk and Eilenberg's 1936 problem of classifying, up to homotopy, the maps from a solenoid complement $S^3\backslash\Sigma$ to the $2$-sphere, is essentially hyperfinite but not smooth. In the course of this work, we record Borel definable versions of a number of classical results bearing on both the combinatorial and homotopical formulations of \v{C}ech cohomology; in aggregate, this work may be regarded as laying foundations for the descriptive set theoretic study of the homotopy relation on the space of maps from a locally compact Polish space to a polyhedron, a relation which embodies a substantial variety of classification problems arising throughout mathematics., Comment: 63 pages; revised and accepted to Forum of Mathematics, Pi

We present a new aspect of the study of higher derived limits. More precisely, we introduce a complexity measure for the elements of higher derived limits over the directed set $\Omega$ of functions from $\mathbb{N}$ to $\mathbb{N}$ and prove that cocycles of this complexity are images of cochains of the roughly the same complexity. In the course of this work, we isolate a partition principle for powers of directed sets and show that whenever this principle holds, the corresponding derived limit $\mathrm{lim}^n$ is additive; vanishing results for this limit are the typical corollary. The formulation of this partition hypothesis synthesizes and clarifies several recent advances in this area., Comment: Accepted for publication in the Journal of the European Mathematical Society

We further develop a forcing notion known as Coding with Perfect Trees and show that this poset preserves, in a strong sense, definable $P$-points, definable tight MAD families and definable selective independent families. As a result, we obtain a model in which $\mathfrak{a}=\mathfrak{u}=\mathfrak{i}=\aleph_1<2^{\aleph_0}=\aleph_2$, each of $\mathfrak{a}$, $\mathfrak{u}$, $\mathfrak{i}$ has a $\Pi^1_1$ witness and there is a $\Delta^1_3$ well-order of the reals. Note that both the complexity of the witnesses of the above combinatorial cardinal characteristics, as well as the complexity of the well-order are optimal. In addition, we show that the existence of a $\Delta^1_3$ well-order of the reals is consistent with $\mathfrak{c}=\aleph_2$ and each of the following: $\mathfrak{a}=\mathfrak{u}<\mathfrak{i}$, $\mathfrak{a}=\mathfrak{i}<\mathfrak{u}$, $\mathfrak{a}<\mathfrak{u}=\mathfrak{i}$, where the smaller cardinal characteristics have co-analytic witnesses. Our methods allow the preservation of only sufficiently definable witnesses, which significantly differs from other preservation results of this type., Comment: 19 pages

A question dating to Sibe Marde\v{s}i\'{c} and Andrei Prasolov's 1988 work Strong homology is not additive, and motivating a considerable amount of set theoretic work in the ensuing years, is that of whether it is consistent with the ZFC axioms for the higher derived limits $\mathrm{lim}^n$ $(n>0)$ of a certain inverse system $\mathbf{A}$ indexed by ${^\omega}\omega$ to simultaneously vanish. An equivalent formulation of this question is that of whether it is consistent for all $n$-coherent families of functions indexed by ${^\omega}\omega$ to be trivial. In this paper, we prove that, in any forcing extension given by adjoining $\beth_\omega$-many Cohen reals, $\mathrm{lim}^n \mathbf{A}$ vanishes for all $n > 0$. Our proof involves a detailed combinatorial analysis of the forcing extension and repeated applications of higher dimensional $\Delta$-system lemmas. This work removes all large cardinal hypotheses from the main result of arXiv:1907.11744 and substantially reduces the least value of the continuum known to be compatible with the simultaneous vanishing of $\mathrm{lim}^n \mathbf{A}$ for all $n > 0$., Comment: 30 pages, 1 figure

We show that it is consistent relative to a weakly compact cardinal that strong homology is additive and compactly supported within the class of locally compact separable metric spaces. This complements work of Marde\v{s}i\'{c} and Prasolov showing that the Continuum Hypothesis implies that a countable sum of Hawaiian earrings witnesses the failure of strong homology to possess either of these properties. Our results build directly on work of Lambie-Hanson and the second author which establishes the consistency, relative to a weakly compact cardinal, of $\mathrm{lim}^s \mathbf{A} = 0$ for all $s \geq 1$ for a certain pro-abelian group $\mathbf{A}$; we show that that work's arguments carry implications for the vanishing and additivity of the $\mathrm{lim}^s$ functors over a substantially more general class of pro-abelian groups indexed by $\mathbb{N}^{\mathbb{N}}$., Comment: 17 pages; expanded exposition. Comments welcome

This is the first installment in a series of papers in which we illustrate how classical invariants of homological algebra and algebraic topology can be enriched with additional descriptive set-theoretic information. To effect this enrichment, we show that many of these invariants can be naturally regarded as functors to the category, introduced herein, of groups with a Polish cover. The resulting definable invariants provide far stronger means of classification. In the present work we focus on the first derived functors of $\mathrm{Hom}(-,-)$ and $\mathrm{lim}(-)$. The resulting definable $\mathrm{Ext}(B,F)$ for pairs of countable abelian groups $B,F$ and definable $\mathrm{lim}^{1}(\boldsymbol{A})$ for towers $\boldsymbol{A}$ of Polish abelian groups substantially refine their classical counterparts. We show, for example, that the definable $\textrm{Ext}(-,\mathbb{Z})$ is a fully faithful contravariant functor from the category of finite rank torsion-free abelian groups $\Lambda$ with no free summands; this contrasts with the fact that there are uncountably many non-isomorphic such groups $\Lambda$ with isomorphic classical invariants $\textrm{Ext}(\Lambda,\mathbb{Z}) $. To facilitate our analysis, we introduce a general Ulam stability framework for groups with a Polish cover and we prove several rigidity results for non-Archimedean abelian groups with a Polish cover. A special case of our main result answers a question of Kanovei and Reeken regarding quotients of the $p$-adic groups. Finally, using cocycle superrigidity methods for profinite actions of property (T) groups, we obtain a hierarchy of complexity degrees for the problem $\mathcal{R}(\mathrm{Aut}(\Lambda)\curvearrowright\mathrm{Ext}(\Lambda,\mathbb{Z}))$ of classifying all group extensions of $\Lambda$ by $\mathbb{Z}$ up to base-free isomorphism, when $\Lambda =\mathbb{Z}[1/p]^{d}$ for prime numbers $p$ and $ d\geq 1$., Comment: Minor revisions; 38 pages

The point of departure for the present work is Barry Mitchell's 1972 theorem that the cohomological dimension of $\aleph_n$ is $n+1$. We record a new proof and mild strengthening of this theorem; our more fundamental aim, though, is some clarification of the higher-dimensional infinitary combinatorics lying at its core. In the course of this work, we describe simplicial characterizations of the ordinals $\omega_n$, higher-dimensional generalizations of coherent Aronszajn trees, bases for critical inverse systems over large index sets, nontrivial $n$-coherent families of functions, and higher-dimensional generalizations of portions of Todorcevic's walks technique. These constructions and arguments are undertaken entirely within a $\mathsf{ZFC}$ framework; at their heart is a simple, finitely iterable technique of compounding $C$-sequences., Comment: Accepted to Advances in Mathematics. 61 pages; a substantial addition of details to many of the previous version's arguments. Comments are welcome

We show that the classical homology theory of Steenrod may be enriched with descriptive set-theoretic information. We prove that the resulting definable homology theory provides a strictly finer invariant than Steenrod homology for compact metrizable spaces up to homotopy. In particular, we show that pro-tori are completely classified up to homeomorphism by their definable homology. This is in contrast with the fact that, for example, there exist uncountably many pairwise non-homeomorphic solenoids with the same Steenrod homology groups. We similarly develop a definable cohomology theory which strengthens \v{C}ech cohomology and we show that it completely classifies complements of pro-tori up to homeomorphism. We also apply definable cohomology theory to the study of the space $\left[ X,S^{2}\right] $ of homotopy classes of continuous functions from a solenoid complement $X$ to the $2$-sphere, which was initiated by Borsuk and Eilenberg in 1936. It was proved by Eilenberg and Steenrod in 1940 that the space $\left[ X,S^{2}\right] $ is uncountable. We will strengthen this result, by showing that each orbit of the canonical action $\mathrm{Homeo}% \left( X\right) \curvearrowright \left[ X,S^{2}\right] $ is countable, and hence that such an action has uncountably many orbits. This can be seen as a rigidity result, and will be deduced from a rigidity result for definable automorphisms of the \v{C}ech cohomology of $X$. We will also show that these results still hold if one replaces solenoids with pro-tori. We conclude by applying the machinery developed herein to bound the Borel complexity of several well-studied classification problems in mathematics, such as that of automorphisms of continuous-trace $C^{*}$-algebras up to unitary equivalence, or that of Hermitian line bundles, up to isomorphism, over a locally compact second countable space., Comment: These notes provide an early draft of what has now been corrected and expanded to the series of papers "The Definable Content of Homological Invariants I, II, and III" (see arXiv:2008.08782 and arXiv:2210.11098, for the first two papers in the series).The reader is strongly advised to refer to the series for a more detailed and rigorous treatment

In 1988, Sibe Marde\v{s}i\'{c} and Andrei Prasolov isolated an inverse system $\mathbf{A}$ with the property that the additivity of strong homology on any class of spaces which includes the closed subsets of Euclidean space would entail that $\lim^n\mathbf{A}$ (the $n^{\text{th}}$ derived limit of $\mathbf{A}$) vanishes for every $n >0$. Since that time, the question of whether it is consistent with the $\mathsf{ZFC}$ axioms that $\lim^n \mathbf{A}=0$ for every $n >0$ has remained open. It remains possible as well that this condition in fact implies that strong homology is additive on the category of metric spaces. We show that, assuming the existence of a weakly compact cardinal, it is indeed consistent with the $\mathsf{ZFC}$ axioms that $\lim^n \mathbf{A}=0$ for all $n >0$. We show this via a finite support iteration of Hechler forcings which is of weakly compact length. More precisely, we show that in any forcing extension by this iteration a condition equivalent to $\lim^n\mathbf{A}=0$ will hold for each $n>0$. This condition is of interest in its own right; namely, it is the triviality of every coherent $n$-dimensional family of certain specified sorts of partial functions $\mathbb{N}^2\to\mathbb{Z}$ which are indexed in turn by $n$-tuples of functions $f:\mathbb{N}\to\mathbb{N}$. The triviality and coherence in question here generalize the well-studied case of $n=1$., Comment: 35 pages. Accepted to Forum of Mathematics: Pi

We define well-connectedness, an order-theoretic notion of largeness whose associated partition relations $\nu\to_{wc}(\mu)_\lambda^2$ formally weaken those of the classical Ramsey relations $\nu\to(\mu)_\lambda^2$. We show that it is consistent that the arrows $\to_{wc}$ and $\to$ are, in infinite contexts, essentially indistinguishable. We then show, in contrast, that in Mitchell's model of the tree property at $\omega_2$, the relation $\omega_2\to_{wc}(\omega_2)_\omega^2$ does hold, and that the consistency strength of this relation holding is precisely a weakly compact cardinal. These investigations may be viewed as augmenting those of [BHS], the central arrow of which, $\to_{hc}$, is of intermediate strength between $\to_{wc}$ and the Ramsey arrow $\to$., Comment: 8 pages

In this paper, the first in a projected two-part series, we describe an organizing framework for the study of infinitary combinatorics. This framework is \v{C}ech cohomology. We show in particular that the \v{C}ech cohomology groups of the ordinals articulate higher-dimensional generalizations of Todorcevic's walks and coherent sequences techniques, and begin to account for those techniques' `unreasonable effectiveness' on $\omega_1$. This discussion occupies the first half of our paper and is written with a general mathematical audience in mind. We turn in the paper's second half to more properly set-theoretic considerations. We describe a number of consistency results on the cohomology groups of the ordinals which certify their status as a graded family of incompactness principles. We show in particular that nontrivial cohomology groups on the ordinals are in some tension with large cardinals, and are maximally extant in G\"{o}del's model $\mathrm{L}$. We describe forcings to add, then trivialize, nontrivial $n$-cocycles, and conclude with some comparison of these principles with those benchmark incompactness phenomena, the existence of square sequences and failures of stationary reflection., Comment: 45 pages; revision of Lemma 2.22 (and several preparatory remarks) occasioned version 2

An infinite graph is highly connected if the complement of any subgraph of smaller size is connected. We consider weaker versions of Ramsey's Theorem asserting that in any coloring of the edges of a complete graph there exist large highly connected subgraphs all of whose edges are colored by the same color., Comment: 11 pages

We consider the question of the additivity of strong homology. This entails isolating the set-theoretic content of the higher derived limits of an inverse system indexed by the functions from $\mathbb{N}$ to $\mathbb{N}$. We show that this system governs, at a certain level, the additivity of strong homology over sums of arbitrary cardinality. We show in addition that, under the Proper Forcing Axiom, strong homology is not additive, not even on closed subspaces of $\mathbb{R}^4$.