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The Wallace problem and countably compact torsion-free Abelian groups in ZFC

This paper proves in ZFC that every torsion-free Abelian group of cardinality $\mathfrak c$ admits a Hausdorff countably compact group topology without nontrivial convergent sequences, thereby providing a negative answer to Wallace's question by constructing a commutative Tychonoff countably compact topological semigroup with two-sided cancellation that is not a group.

Original authors: Juliane Trianon Fraga, Vinicius de Oliveira Rodrigues

Published 2026-08-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Juliane Trianon Fraga, Vinicius de Oliveira Rodrigues

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of mathematics, there is a vast landscape of shapes and structures that behave like numbers but follow their own internal rules. Among these are "groups," collections of objects that can be combined in a specific way, much like adding numbers, but where the objects themselves might be complex patterns or infinite lists. When these groups are given a "topology," they gain a sense of closeness and distance, allowing mathematicians to talk about sequences of objects getting closer and closer to a specific point, eventually arriving there. This blend of algebra and geometry creates "topological groups," which are central to understanding symmetry and continuity in the universe. A particularly stubborn puzzle in this field has concerned "countably compact" groups. These are structures where any infinite list of points must have a cluster of points nearby, ensuring the space never drifts off into chaos. For decades, mathematicians wondered if a specific type of these groups, which have no repeating cycles and allow for perfect cancellation of operations, could exist without any sequence of points actually converging to a limit. If such a group existed, it would break a long-standing assumption about how these mathematical worlds must behave.

For nearly seventy-five years, a question posed by a mathematician named A. D. Wallace remained unanswered. He asked whether a specific kind of mathematical structure, known as a semigroup, that is compact enough to keep points from wandering off and allows for perfect cancellation of operations, must necessarily be a full-fledged group. In simpler terms, if you have a system where you can combine elements and undo those combinations perfectly, and the system is tightly packed, does it automatically have to be a group? The answer was known to be "yes" if the system was perfectly compact, but no one knew if the slightly weaker condition of "countable compactness" was enough to force the same result. Previous attempts to find a counterexample required assuming extra, unproven rules about the nature of infinity, leaving the question open in the standard rules of mathematics.

A team of researchers has now solved this problem using only the standard rules of mathematics, without needing any extra assumptions. They proved that such a counterexample does exist. Specifically, they constructed a mathematical object that behaves like a group in almost every way—it is torsion-free, meaning no element repeats in a cycle, and it is countably compact, meaning it is tightly packed. However, it is not a group because it lacks a crucial property: it contains no non-trivial sequences that converge to a limit. In this structure, you can list an infinite number of distinct points, and they will never settle down to a single destination, no matter how you look at them. This discovery confirms that the answer to Wallace's question is "no." A system can be tightly packed and allow for perfect cancellation without being a full group.

The researchers achieved this by building a massive, infinite collection of numbers and defining a very specific way to measure distance between them. They started with a free Abelian group, which is essentially a collection of vectors with integer coordinates, and carefully crafted a topology, or a rule for closeness, that prevents any sequence from converging unless it eventually stops changing. They used a technique involving "ultrafilters," which are sophisticated tools for deciding which infinite sets of numbers are "large" enough to matter, to ensure that every possible infinite list of points has an accumulation point nearby, satisfying the compactness requirement. Yet, they simultaneously ensured that no list of distinct points could actually reach a limit, preserving the "no convergent sequences" property. This delicate balancing act was performed entirely within the standard framework of mathematics, proving that the existence of such a structure is a fundamental fact, not a possibility that depends on extra hypotheses.

The implications of this construction ripple through several other areas of mathematics. Because the object they built is a group with these specific properties, it can be used to create other structures that were previously only known to exist under uncertain conditions. For instance, the researchers showed that this group contains a substructure that acts as a "Wallace semigroup," a commutative system with two-sided cancellation that is countably compact but not a group. This settles a debate that had persisted for decades. Furthermore, their work provides a concrete example of a "paratopological group," a structure where the operation of combining elements is continuous, but the reverse operation is not. This answers questions about whether such imperfect groups can be tightly packed. They also demonstrated that this group can be used to build a "monothetic" monoid, a system generated by a single element, which is countably compact but not a group.

The paper also addresses a question about the size of these mathematical spaces. The researchers proved that in the specific group they constructed, any infinite closed set of points must be as large as the entire continuum of real numbers. This means there are no "small" infinite clusters hidden inside; if a set is infinite and closed, it is maximally large. This result resolves a specific inquiry about the density of points in such groups. By constructing this object, the authors have not only answered a famous question but have also provided a versatile tool that generates solutions to several other open problems in topology and algebra. Their work stands as a definitive proof that the mathematical universe contains these elusive, tightly packed structures that defy the intuition that tightness and cancellation must always lead to a group. The existence of these objects is now a settled fact, grounded in the standard axioms of mathematics, changing the landscape of what is known about the behavior of infinite algebraic systems.

Technical Summary: The Wallace Problem and Countably Compact Torsion-Free Abelian Groups in ZFC

1. The Problem and Context
The paper addresses two interconnected, long-standing problems in topological algebra and set-theoretic topology, both of which had remained open in Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC) for decades.

The Wallace Problem: In 1955, A. D. Wallace asked whether every Hausdorff countably compact topological semigroup with two-sided cancellation must be a topological group. A counterexample is known as a "Wallace semigroup." While consistent constructions of such semigroups existed under additional set-theoretic hypotheses (e.g., the Continuum Hypothesis or Martin's Axiom), the existence of a Wallace semigroup in ZFC remained an open question for 73 years.
The Existence of Non-Torsion Countably Compact Groups: A related problem, posed by van Douwen and Comfort, concerns the existence of a countably compact topological group without nontrivial convergent sequences that is not a torsion group. Prior to this work, ZFC constructions of countably compact groups without nontrivial convergent sequences were limited to torsion groups. Specifically, it was unknown whether the free Abelian group on $\mathfrak{c}$ generators, $Z(\mathfrak{c})$ , or the additive group of real numbers, $\mathbb{R}$ , admitted such topologies in ZFC.

2. Methodology
The authors construct the required topologies using a combination of algebraic embedding, uniform finite approximation, and a transfinite recursive construction involving "block ultrafilters."

Algebraic Reduction: The authors first establish that any torsion-free Abelian group $G$ of cardinality $\mathfrak{c}$ can be embedded as a subgroup of the direct sum $Q(\mathfrak{c})$ such that $Z(\mathfrak{c}) \leq G \leq Q(\mathfrak{c})$ . This allows them to treat the problem within the context of vector spaces over $\mathbb{Q}$ with a specific basis structure.
Uniform Finite Kronecker Lemma: A central technical tool is a uniform finite version of Kronecker's theorem (Lemma 3.1). This lemma ensures that for any finite set of elements in an Abelian group and any target values in the circle group $\mathbb{T}$ , one can find a homomorphism to $\mathbb{T}$ that approximates these targets, provided the targets satisfy certain integer relations with bounded coefficients. This relies on the Stone–Weierstrass theorem and Arzelà–Ascoli compactness arguments.
Bounded Deletion and Independence: To handle the infinite construction, the authors employ a combinatorial deletion argument (Lemma 4.4). Given a finite set $A$ and a large set $X$ , they identify a large subset $Y \subseteq X$ that is "independent" of $A$ regarding bounded integer relations. This allows the construction of homomorphisms that are small on $Y$ while preserving values on $A$ .
Block Ultrafilters and Transfinite Recursion: The construction proceeds by enumerating all injective sequences in $G$ as $(s_\alpha)_{\alpha < \mathfrak{c}}$ . For each sequence, the authors define a "block" structure on the natural numbers $\mathbb{N}$ and associate a free ultrafilter $p_\alpha$ . They recursively construct a sequence of homomorphisms $\psi_l: G \to \mathbb{T}$ that converge pointwise to a limit homomorphism $\psi$ . The construction ensures that for every non-zero $x \in G$ , there is a homomorphism $\psi_x$ such that $\psi_x(x) \neq 0$ , and for every sequence $s_\alpha$ , the ultrafilter limit of the sequence under the constructed topology converges to a specific non-zero element (specifically, a basis vector $e_{\iota(\alpha)}$ ). The final topology $\tau_G$ is the initial topology induced by the family of these homomorphisms $\{\psi_x : x \in G \setminus \{0\}\}$ mapping $G$ into the product space $\mathbb{T}^{G \setminus \{0\}}$ .

3. Key Contributions and Results
The primary contribution is the resolution of the existence questions in ZFC without additional set-theoretic assumptions.

Theorem 1.1: In ZFC, every torsion-free Abelian group of cardinality $\mathfrak{c}$ $c$ admits a Hausdorff countably compact group topology in which every convergent sequence is eventually constant (i.e., there are no nontrivial convergent sequences).
- This resolves the Tkachenko problem for the free Abelian group $Z(\mathfrak{c})$ .
- It resolves the question for the Baer–Specker group $Z^\omega$ .
- It resolves the question for the additive group of real numbers $\mathbb{R}$ (via the isomorphism $Q(\mathfrak{c}) \cong \mathbb{R}$ ).
Corollary 1.3 (The Wallace Semigroup): By specializing the construction to the non-negative cone $P$ of $Z(\mathfrak{c})$ , the authors prove the existence in ZFC of a commutative Tychonoff countably compact topological semigroup with two-sided cancellation that is not a group. This provides a negative answer to Wallace's 1955 question.
Further Consequences: The paper demonstrates that the main theorem implies several other results previously known only under additional hypotheses:
- The existence in ZFC of a Tychonoff countably compact topological semigroup containing a copy of the bicyclic semigroup.
- The existence in ZFC of a functionally Hausdorff countably compact paratopological group that is not a topological group (answering Guran's questions).
- The existence in ZFC of a Hausdorff countably compact monothetic topological monoid that is a submonoid of a topological group but not a group.
- The existence of a countably compact free Abelian group where every infinite closed subset has cardinality at least $\mathfrak{c}$ .

4. Significance
The paper claims to settle a 73-year-old open problem regarding the Wallace semigroup entirely within ZFC. Prior to this work, the existence of such semigroups was known only under stronger set-theoretic axioms. By constructing a torsion-free countably compact group without nontrivial convergent sequences in ZFC, the authors bridge the gap between topological group theory and semigroup theory, showing that the "Wallace phenomenon" is not an artifact of specific set-theoretic models but a feature of ZFC itself. The work also unifies and resolves multiple related questions concerning the structure of countably compact groups, paratopological groups, and suitable sets, demonstrating that the obstruction to these structures existing in ZFC was not inherent to the axioms but to the lack of a specific construction technique.

The authors explicitly note that their construction relies on the injectivity of the circle group $\mathbb{T}$ in the category of Abelian groups and a careful interplay between combinatorial deletion arguments and analytic approximation via trigonometric polynomials. The result is presented as a definitive solution to the stated problems, with no reliance on the Continuum Hypothesis or Martin's Axiom.

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