The Eilenberg-Ganea conjecture, in algebraic topology, was formulated by Samuel Eilenberg and Tudor Ganea in 1957. It states that a group of cohomological dimension two has a two-dimensional Eilenberg-MacLane space; for every other dimension n, a group of cohomological dimension n is already known to have an n-dimensional Eilenberg-MacLane space, and a group of cohomological dimension two is known to have at least a three-dimensional one. In 1997, Mladen Bestvina and Noel Brady constructed a group that is either a counterexample to the Eilenberg-Ganea conjecture or a counterexample to the related Whitehead conjecture. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementIf a group G has cohomological dimension 2, then it has a 2-dimensional Eilenberg-MacLane space K(G,1). 1 Proposed Year Progress Toward ResolutionIn 1997 Bestvina and Brady constructed a group that either counterexamples this conjecture or the Whitehead conjecture, so both cannot be true; still open. 1 Classification
Resolution Status Prize Status
Prize Status (category) Connections
In Branch
Source Eilenberg-Ganea conjecture (Wikipedia)
Sources
1. Eilenberg-Ganea Conjecture (Wikipedia)
Wikimedia FoundationLead section
It was formulated by Samuel Eilenberg and Tudor Ganea in 1957, in a short but influential paper.
Lead section, first paragraph
if a group G has cohomological dimension 2, then it has a 2-dimensional Eilenberg-MacLane space
Lead section, second paragraph
it is not possible for both conjectures to be true
View the Source Eilenberg-Ganea conjecture (Wikipedia)
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