Eilenberg-Mazer Swindle Overview

• Eilenberg-Mazer swindle is a collection of geometric, algebraic, and categorical techniques that use infinite processes to prove results like vanishing homology and duality theorems.
• It leverages methods such as attaching infinite tails and reindexing panels in products of trees to cancel out lower-dimensional homology and characterize amenability in large-scale geometry.
• The swindle also underpins algebraic axiomatizations of infinite summation, influencing constructions in aperiodic tilings, index theory, and the structure of infinite sum operations.

The Eilenberg-Mazer Swindle refers to a collection of geometric, algebraic, and categorical techniques that exploit infinite processes—often involving the systematic rearrangement or "swindling" of certain structures—to establish results about homology, infinite summation, or duality theorems in algebra. Originating from the classical Eilenberg swindle and extended through contributions by Barry Mazur and others, these techniques play a foundational role in areas such as large scale geometric topology, the duality theory of algebra and languages, and the algebraic axiomatization of infinite sums.

1. Classical and Higher-Dimensional Eilenberg Swindles

The classical Eilenberg swindle is a geometric construction that allows for the vanishing of certain homology classes by leveraging infinite "tails" within a graph or complex. Originally, the Eilenberg swindle attached an infinite tail—a sequence of 1-simplices escaping to infinity—to a given vertex in a (potentially infinite) graph or space. This operation is formalized as

$t_x = [x, x_1] + [x_1, x_2] + [x_2, x_3] + \cdots$

where each $[x_k, x_{k+1}]$ is a 1-simplex and the points $x_k$ diverge in the space.

A significant extension of this idea has been developed for products of trees and, more generally, higher-dimensional complexes. In this context, the technique systematically constructs, for any $k$-cycle $c$ in a product of trees, $(k+1)$-chains (“panels”) attached along various coordinate hyperplanes. For a fixed cube $Q$ and direction $j$, one forms

$$T_j = \sum_Q c(Q) \cdot (\text{panel attached to } Q)$$

and then modifies the original cycle as

$c' = c - \partial T_j,$

ensuring that the coefficients on all $k$-cubes in the chosen hyperplane vanish. Iterating this process across all directions in a product of $n$ trees kills off all homology classes in degrees $k < n$, with the top degree remaining infinite dimensional. This is the content of:

$$H_k^{\mathrm{uf}}(T_1 \times \cdots \times T_n; R) = 0 \qquad (k \leq n - 1)$$

$$H_n^{\mathrm{uf}}(T_1 \times \cdots \times T_n; R) \text{ is infinite dimensional}$$

where $H_k^{\mathrm{uf}}$ denotes uniformly finite homology, a large scale homology theory (Diana et al., 2014).

2. Algebraic and Categorical Swindling: Eilenberg-Type Correspondence

Within the framework of general algebra, the Eilenberg-Mazer swindle reappears in the form of categorical duality arguments—so-called "Eilenberg-type correspondences." Here, the central mechanism is to transfer algebraic description (in terms of varieties or pseudovarieties of algebras, governed by Birkhoff's theorem and its finite-algebra analog) to their coalgebraic (or "language-theoretic") duals.

Given a monad $T$ on a category $\mathcal{D}$ (the "algebraic" side), one may dualize—using a contravariant adjunction—to obtain a comonad $B$ on a dual category $\mathcal{C}$ (the "coalgebraic" or "language" side). Equational $T$-theories (families of morphisms $T X \rightarrow Q_X$) correspond under this duality to coequational $B$-theories. The Eilenberg-type correspondence is then:

$$\text{Varieties or pseudovarieties of } T\text{-algebras} \quad \longleftrightarrow \quad \text{(Co)equational } B\text{-theories}$$

This bijection is the abstract underpinning of results such as Eilenberg's variety theorem for regular languages and its generalizations (Salamanca, 2017).

A plausible implication is that, much as the geometric swindle “kills” cycles, the categorical swindle “transforms” equational content into dual observational (language-theoretic) closure properties, thereby unifying previously disparate classifications across algebra and language theory.

3. The Swindle and Infinite Summation Axoms

The Eilenberg-Mazer swindle is also central to the algebraic axiomatization of infinite summation. When attempting to "infinitize" associativity or distributivity in an algebraic setting (as opposed to a topological one), the process of reindexing, rearranging, or swindling terms leads to highly restrictive conditions.

A key example is "prefix associativity," where, for any summable sequence $(a_0,a_1,...)$, one expects

$a_0 + (a_1 + a_2 + ...) = \sum_i a_i.$

In the presence of right cancellation (e.g., in groups), the Eilenberg–Mazur swindle applied to constant sequences establishes that only the trivial element can survive if one attempts to maintain such infinite associativity (Theorem 3.9 in (Nielsen, 19 Aug 2025)). Thus, enforcing unrestricted infinite associativity leads to degenerate structure, unless extra conditions or new axioms are imposed.

Further, insertive associativity and "left reorderability" (Axiom 7.5 in (Nielsen, 19 Aug 2025)) dictate that, under certain reindexing and multiplication actions, double sums may be reassociated:

$$\sum_{i \in I}\left[ \sum_{k \in \psi'(i)} r_k \right] a_i = \sum_{k \in K} r_k \left[ \sum_{i \in \psi(k)} a_i \right].$$

With mild additional hypotheses (singleton sums, normal behavior of $1$ and $-1$), this is sufficient to recover a (possibly degenerate) ring structure from the summation axioms.

4. Applications in Large Scale Geometry and Group Theory

The higher-dimensional Eilenberg swindle has direct applications in large scale geometry and geometric group theory. The technique is crucial in computing uniformly finite homology for products of trees and analyzing group homology with $\ell_\infty$ coefficients for lattices in such products. Specifically, for a lattice $\Gamma$ acting properly and cocompactly on $T_1 \times \cdots \times T_n$ (with each $T_i$ an infinite tree),

$$H_k^{\mathrm{uf}}(\Gamma; R) = \begin{cases} 0 & k \neq n \ \text{infinite dimensional} & k = n \end{cases}$$

The vanishing in lower dimensions reflects the non-amenability of the product space, while the infinite dimensionality in top degree encodes large scale geometric complexity (Diana et al., 2014).

One notable consequence is a refined characterization of amenability: for a finitely generated group $T$ and an infinite tree $T$, the product $\mathrm{Cay}(T) \times T$ has non-vanishing 1-homology if and only if $T$ is amenable (Corollary 4 in (Diana et al., 2014)). This relates amenability, a classic notion in group theory, to vanishing properties of large scale homology in higher degrees.

5. Implications for Aperiodic Tilings and Index Theory

Techniques derived from the Eilenberg-Mazer swindle serve as the foundation for constructing aperiodic tilings using higher homological data. Notably, by employing almost equivariant homology—chains with only finitely many values—alongside high-dimensional swindle arguments, one proves that certain transfer maps from ordinary to almost-equivariant homology fail to surject. For a compact manifold $M \times N$ and universal cover $\widetilde{M \times N}$, the non-vanishing of $H_{1}(M \times N; R)$ and the vanishing of $H_1^{\mathrm{ae}}(\widetilde{M \times N}; R)$ due to the swindle guarantee aperiodicity in the resulting tiling. No cocompact symmetry group can exist, as the homological “decoration” cannot be canceled finitely (Diana et al., 2014).

Uniformly finite homology with coefficients in $\ell_\infty$ thus also becomes an essential invariant in index theory and rigidity phenomena. The swindle method’s ability to "kill" certain classes is central to both purely mathematical and applied contexts.

6. Maximality, Quotients, and Topology of Infinite Sums

In algebraic frameworks for infinite summation, the Eilenberg-Mazer swindle reveals the limits of possible axiomatizations. If unconditional (absolute) summation is extended while preserving distributivity and related axioms, the resulting system is maximal—no further extension is possible (Proposition 9.4 in (Nielsen, 19 Aug 2025)). The theory provides criteria for when quotient constructions preserve infinite summation; specifically, a subgroup $S$ must be $\Sigma$-closed (every summable family in $S$ sums to an element of $S$) for quotient summation systems to be well defined (Proposition 10.2 in (Nielsen, 19 Aug 2025)).

Additionally, knowledge of the infinite sums determines a canonical "Σ-topology," which refines the original topology and sometimes coincides with the classical Hausdorff topology.

7. Synthesis and Broader Significance

The Eilenberg-Mazer swindle—whether in the classical geometric sense, categorical duality, or algebraic axiomatization—encapsulates a recurring principle: that infinite processes, rearrangements, and reindexings, if handled without sufficient constraints, force vanishing or degeneracy in the structures under consideration (homology, ring axioms, variety classification). When appropriately controlled by new axioms, duality frameworks, or panel constructions, the swindle delivers powerful tools for explicit computation, characterization of structural properties (such as amenability), and the construction of exotic examples (such as aperiodic tilings).

This principle provides a conceptual bridge across large scale geometry, algebra, category theory, and mathematical physics, substantiating a unifying perspective on the behavior of infinite systems under algebraic and geometric operations.