A group whose algebra needs two dimensions but whose space needs three

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Family 249 · Group theory
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-22

1. The question

A group can be studied through a space whose loops record its multiplication. The space may use vertices, edges, filled triangles and higher-dimensional cells. How many dimensions are really necessary?

There are two ways to ask. One is geometric: build an actual classifying space for the group and minimize its dimension. The other is algebraic: replace the space by a chain of modules recording its homological information and minimize the length of that chain.

The algebraic answer cannot exceed the geometric one. For most dimensions, classical theorems say that the answers agree. Dimension two is the exceptional case, and the Eilenberg–Ganea conjecture asks whether equality holds there as well.

Family 249 claims a counterexample: a finitely generated residually finite group whose integral cohomological dimension is two but whose geometric dimension is three. The source rules out every two-dimensional classifying space, including spaces with infinitely many cells. Its group is not finitely presented, a distinction that matters when interpreting the result.

2. Background and essential definitions

Spaces assembled from cells

A CW complex is a space built by attaching cells: points, intervals, disks, three-dimensional balls and their higher-dimensional analogues. Its dimension is the largest dimension of a cell used. A complex can have infinitely many cells while still being two-dimensional.

The fundamental group $\pi_1(X)$ records loops in a connected space $X$, with loops identified when one can be continuously deformed into the other. A classifying space $K(G,1)$ has fundamental group $G$ and contractible universal cover.

Contractible means that the entire space can be continuously collapsed to a point. The universal cover removes the loops encoded by the fundamental group; requiring it to be contractible ensures that no additional higher homotopy remains.

The geometric dimension $\operatorname{gd}G$ is the least dimension of a classifying space for $G$. A circle is a one-dimensional classifying space for $\mathbb Z$, because its universal cover is a line. More generally, a bouquet of circles gives a one-dimensional model for a free group.

The algebraic dimension

The group ring $\mathbb Z[G]$ consists of finite integer linear combinations of group elements, with multiplication extended from the group. Modules over this ring are abelian groups equipped with compatible $G$-actions.

A projective resolution of the trivial module $\mathbb Z$ is an exact sequence of projective modules ending in $\mathbb Z$. Exactness means that the output of one map is precisely the part killed by the next. Projective modules behave like direct summands of free modules and are the natural building blocks for this algebraic problem.

The integral cohomological dimension $\operatorname{cd}_{\mathbb Z}G$ is the shortest possible resolution length. For a classifying space, the cellular chains of its universal cover provide such a resolution. Consequently,

$$\operatorname{cd}_{\mathbb Z}G\leq\operatorname{gd}G.$$

The issue is whether a short algebraic resolution can always be realized by an equally low-dimensional contractible covering space.

Acyclic does not mean contractible

A connected space is acyclic over the integers when all its reduced integral homology groups vanish. Homology measures cycles modulo boundaries using additive algebra. It forgets some nonlinear information about how paths are composed.

A contractible space is acyclic, but an acyclic space can still have nontrivial fundamental group. This is the crucial gap. A two-dimensional acyclic complex may supply the desired short algebraic resolution without being the universal cover of a classifying space.

A group is residually finite if each nonidentity element has a nonidentity image in some finite quotient. Finite generation means that finitely many elements generate the group; finite presentation additionally requires finitely many defining relations. The new example has the former property but not the latter.

3. Development history and previous partial results

Eilenberg and Ganea proved in 1957 that geometric and integral cohomological dimensions agree when the latter is at least three. In cohomological dimension one, Stallings proved freeness under finite generation, and Swan removed that restriction. Free groups have one-dimensional classifying spaces.

The unresolved case was therefore sharply localized: could cohomological dimension two force geometric dimension three?

Bestvina and Brady’s 1997 work produced a rich supply of candidates using kernels of height homomorphisms on right-angled Artin groups. Their examples separated homological finiteness conditions from geometric ones. For suitable flag complexes related to a spine of the Poincaré homology sphere, they proved that at least one of the Eilenberg–Ganea and Whitehead conjectures must fail.

Whitehead’s asphericity conjecture asks whether every connected subcomplex of an aspherical two-dimensional CW complex is itself aspherical. Aspherical means that the universal cover is contractible. The Bestvina–Brady alternative did not decide which conjecture fails, and did not by itself identify the geometric dimension of a particular height kernel.

Dicks and Leary gave presentations of these kernels using edge generators and power relations associated with cycles. Howie studied related examples through the plus construction and obtained free relation modules. These works established much of the algebraic and geometric setting used by the new manuscript.

A recent comparison by Leary and Petrosyan, cited by the source, relates Coxeter and Bestvina–Brady candidates under an additional conjectural condition. The new paper instead claims a direct obstruction for one specified height kernel.

The central historical difficulty is easy to state: showing that one two-dimensional acyclic model is not contractible does not exclude a completely different two-dimensional classifying space. The source’s proposed advance is a comparison argument that addresses every hypothetical model, regardless of how many cells it has.

4. What the new manuscripts claim

The manuscript dated 23 September 2026 asserts the existence of a finitely generated residually finite group $G$ with

$$\operatorname{cd}_{\mathbb Z}G=2, \qquad \operatorname{gd}G=3.$$

These are ordinary group dimensions with integral coefficients. They are not dimensions defined using a family of allowed stabilizers or a special kind of proper group action.

The source specifies a construction. Start with a finite acyclic flag triangulation $L$ of the presentation complex

$$\langle x,y\mid x^2=y^5,\quad x^2=(xy^{-1})^3\rangle.$$

A simplicial complex is flag if every finite collection of pairwise adjacent vertices spans a simplex. This condition makes its associated right-angled Artin group geometrically well behaved.

With one generator $a_v$ for each vertex $v$ of $L$, define

$$A_L=\langle a_v\mid [a_v,a_w]=1 \text{ whenever }\{v,w\}\text{ is an edge of }L\rangle.$$

Thus adjacent generators commute, and no other relations are imposed. The height homomorphism $\lambda:A_L\to\mathbb Z$ sends each $a_v$ to $1$. The claimed example is the kernel $G=\ker\lambda$.

The paper further supplies a length-two resolution of $\mathbb Z$ by finitely generated free $\mathbb Z[G]$-modules. This is the finiteness condition called type FL, and implies type FP∞, meaning that there is a projective resolution with finitely generated terms in every degree.

Nevertheless, $G$ is not finitely presented. The seed complex has nontrivial fundamental group, so the Bestvina–Brady finite-presentability criterion excludes that property. A finite algebraic resolution must therefore not be mistaken for a finite presentation or a finite classifying space.

The geometric nonexistence statement is unrestricted: no two-dimensional $K(G,1)$ exists, even if arbitrary infinite sets of cells are allowed. A three-dimensional model supplies the upper bound and hence the exact geometric dimension.

5. Why the advance matters

Importance: 5/5, provisional. The claimed result settles the exceptional case in a basic comparison between algebraic and geometric dimensions of groups. It would show that a two-dimensional homological description can fail to have any two-dimensional geometric realization.

The example is finitely generated and residually finite. Its obstruction is therefore compatible with both finite generation and separation by finite quotients, rather than arising solely from a very large uncontrolled group.

The stronger exclusion of models with infinitely many cells matters. Merely ruling out a finite two-dimensional model would leave geometric dimension undecided, because geometric dimension does not impose a finite-cell requirement.

The proposed small-labeling obstruction also illustrates a general phenomenon: additive cycle data may fail to capture the order-sensitive products that determine actual topology.

6. The main idea

Slice a three-dimensional cubical space

The right-angled Artin group $A_L$ has a standard cubical classifying space assembled from coordinate tori. Its universal cover $E$ is contractible and three-dimensional. The height homomorphism extends to a real-valued function on $E$, affine on each cube.

Take the level set

$$X=\lambda^{-1}(0)\subset E.$$

The kernel $G$ acts freely on this two-dimensional complex. The paper establishes that $X$ is acyclic and has finitely many cell orbits. Its cellular chains provide the short free resolution. The quotient $E/G$, meanwhile, is a three-dimensional classifying space.

What remains is to prevent some other two-dimensional space from doing better.

Label edges by small rotations

The source uses $H=\mathrm{SU}(2)$, which can be represented by unit quaternions. It is a compact noncommutative group with a notion of distance from the identity.

Assign an element of $H$ to every oriented edge of $X$, and assign the inverse to the reversed edge. Multiplying labels along a path gives its transport. The product around a closed path is called holonomy.

A labeling is flat on triangles when each triangular boundary has product $1$. Even if every edge label is very close to $1$, triangle-flatness need not make the product around every long closed path trivial. Global topology can survive local flatness.

The paper claims that a hypothetical two-dimensional classifying space would force precisely such global triviality for all sufficiently small labelings of this particular $X$. It then constructs arbitrarily small counterexamples.

Compare chains with actual words

The triangular boundary chains of $X$ form a specified free basis of its cycle module. Chains record signed edge counts but forget traversal order. Group-valued path products retain that order, so equality of chains is weaker than equality of products.

The comparison argument starts from an actual presentation associated with the hypothetical classifying space. It adjusts generators and relators so that distinguished boundary chains match the triangular ones. These triangular words alone need not present $G$; that is why an arbitrary actual presentation must enter.

Near the identity of $\mathrm{SU}(2)$, errors caused by changing multiplication order are quadratic in the sizes of the labels. Exact signed-count identities cancel the linear discrepancies. The resulting estimate provides a barrier keeping selected products near the identity.

To solve the remaining relator equations, the source uses a localized version of the Gerstenhaber–Rothaus word-map degree argument. The localization keeps solutions in the portion of the equation space controlled by that barrier. Residual finiteness handles finite collections of constraints on finite quotients; compactness passes to all constraints. A single finite quotient need not carry the entire final labeling.

Preserve a global loop while shrinking every edge

The seed complex $L$ has a nontrivial representation of its fundamental group into $\mathrm{SU}(2)$. The manuscript constructs maps from the graph of $X$ into $L$ for which individual edge images and triangle boundaries become uniformly small, but certain long closed paths trace prescribed loops exactly.

Pulling back transport from the representation makes every edge label arbitrarily close to the identity. Small triangle images give exactly trivial triangular products. A long path representing a loop detected by the representation keeps nontrivial holonomy.

This conflicts with the claimed consequence of any two-dimensional classifying space. The three-dimensional upper bound then gives $\operatorname{gd}G=3$.

7. Scope, limitations, and open questions

The example is finitely generated but not finitely presented. The manuscript therefore does not establish the same counterexample under finite presentation, nor does it give a finite three-dimensional classifying space for this group.

Type FL and FP∞ are homological finiteness conditions. They do not repair that missing finite presentation, and they should not be confused with having finitely many cells in every dimension of a classifying space.

The theorem uses integral cohomological dimension. Changing coefficients or changing the kind of group action being classified leads to different invariants and potentially different questions.

The conclusion also does not settle Whitehead’s asphericity conjecture. It resolves the Eilenberg–Ganea side of an earlier alternative, without deciding every other conjecture that appeared in that alternative.

More broadly, identifying which groups of cohomological dimension two do admit two-dimensional classifying spaces remains a meaningful structural problem. A counterexample to the universal statement does not provide a classification of the positive cases.

8. Sources and further reading

Eilenberg and Ganea, On the Lusternik–Schnirelmann category of abstract groups, Annals of Mathematics 65 (1957), 517–518, is the foundational dimension theorem. Swan, Groups of cohomological dimension one, completes the one-dimensional case.

Bestvina and Brady, Morse theory and finiteness properties of groups, Inventiones Mathematicae 129 (1997), 445–470, supplies the height-kernel construction and the earlier alternative. Dicks and Leary’s presentations and Howie’s plus-construction work develop the relevant algebraic models.

Gerstenhaber and Rothaus, The solution of sets of equations in groups, is the degree-theoretic predecessor. Hatcher’s Algebraic Topology gives background on CW complexes, covering spaces, homology and homotopy.

Source manuscripts

• M1. OpenAI, A finitely generated counterexample to the Eilenberg–Ganea conjecture (September 23, 2026).

Sources consulted on 22 October 2026 at repository revision 1581d4c. Download the manuscript citations.