1 · The essentials
Two ways of measuring how much space a mathematical group needs can give different answers. The supplied abstract constructs a group for which an algebraic measure gives two, while the smallest possible dimension of a corresponding geometric space is three. The group can be generated from finitely many elements. It also has a property called residual finiteness, which allows distinct elements to be distinguished in finite groups. According to the source, this example refutes the Eilenberg–Ganea conjecture. Allowing infinitely many pieces in the geometric space does not make a two-dimensional version possible. The main result has a Lean formalization, although the supplied material does not specify its exact coverage.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine an assembly whose parts list suggests that two dimensions should suffice, but which cannot actually be assembled without a third dimension. Even allowing an unlimited number of pieces would not remove that obstruction; the group's algebraic and geometric measurements differ in a somewhat similar way.
Why it matters
The example separates two measurements of a group that the Eilenberg–Ganea conjecture predicts should agree in this setting. Its finite generation and residual finiteness are explicit features of the construction. The obstruction concerns every two-dimensional classifying space, not merely those built from finitely many cells.
2 · Going further
The precise result stated in the abstract is the construction of a finitely generated, residually finite group with integral cohomological dimension and geometric dimension . Cohomological dimension is an algebraic dimension measured using group cohomology; geometric dimension is the smallest dimension of a classifying space for the group. The repository summary explicitly states that has no two-dimensional classifying space, even if infinitely many cells are allowed.
The manuscript presents this construction as a counterexample to the Eilenberg–Ganea conjecture. The abstract gives the result but does not describe the group, the construction, or the main steps of the argument, so an account of the proof strategy cannot be extracted from the supplied text. Only the main manuscript is supplied; no companion manuscripts or relationships between them are described. The main result has a Lean formalization, but the available information does not identify which individual claims fall within its scope.
Useful notions
- Groups and generators
- Finite groups and residual finiteness
- Group cohomology
- Classifying spaces and dimension
What was known before / what is new
Before
The source identifies the Eilenberg–Ganea conjecture as the question addressed by this construction. It supplies no account of earlier partial results, earlier examples, or the history of the conjecture.
New
The manuscript states that it constructs a finitely generated, residually finite group with integral cohomological dimension two and geometric dimension three. The repository summary emphasizes that no two-dimensional classifying space exists for this group, even with infinitely many cells. The source presents this as a counterexample to the Eilenberg–Ganea conjecture.
Limits and assumptions
This is an existence result for a group with the stated properties, not a claim that every finitely generated or residually finite group has differing algebraic and geometric dimensions. The coefficient choice is integral, and the dimensions asserted are specifically two and three. The main result has a Lean formalization, but its detailed scope is not supplied, so separate formal verification of every stated property should not be assumed. The abstract provides neither construction details nor an argument outline, and it identifies no remaining open questions.
Mini glossary
- Finitely generated group —
- A group whose elements can all be obtained by combining finitely many chosen elements and their inverses.
- Residually finite —
- A group is residually finite if any two distinct elements can be distinguished by a homomorphism to a finite group.
- Integral cohomological dimension —
- An algebraic measure of a group's dimension defined through group cohomology over the integers.
- Classifying space —
- A connected space with the given group as its fundamental group and with contractible universal cover.
- Geometric dimension —
- The smallest dimension of a classifying space for a group.
Keywords : Eilenberg–Ganea conjecture · Cohomological dimension · Geometric dimension
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Constructs a finitely generated residually finite group with integral cohomological dimension two and geometric dimension three, disproving the Eilenberg–Ganea conjecture. It has no two-dimensional classifying space, even with infinitely many cells.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
A finitely generated counterexample to the Eilenberg–Ganea conjecture
23/09/2026We construct a finitely generated residually finite group of integral cohomological dimension two and geometric dimension three, disproving the Eilenberg–Ganea conjecture.