The Eilenberg–Ganea conjecture is a claim in
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
. It was formulated by
Samuel Eilenberg
Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra.
Early life and education
He was born in Warsaw, Kingdom of Poland to ...
and
Tudor Ganea in 1957, in a short, but influential paper. It states that if a group ''G'' has
cohomological dimension 2, then it has a 2-dimensional
Eilenberg–MacLane space
In mathematics, specifically algebraic topology, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name. ...
. For ''n'' different from 2, a group ''G'' of
cohomological dimension ''n'' has an ''n''-dimensional Eilenberg–MacLane space. It is also known that a group of cohomological dimension 2 has a 3-dimensional Eilenberg−MacLane space.
In 1997,
Mladen Bestvina and Noel Brady constructed a group ''G'' so that either ''G'' is a counterexample to the Eilenberg–Ganea conjecture, or there must be a counterexample to the
Whitehead conjecture; in other words, it is not possible for both conjectures to be true.
References
Conjectures
Theorems in algebraic topology
Unsolved problems in mathematics
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