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 a ...

and

Tudor Ganea Tudor Ganea (October 17, 1922 –August 1971) was a Romanian-American mathematician, known for his work in algebraic topology, especially homotopy theory. Ganea left Communist Romania to settle in the United States in the early 1960s. He tau ...

in 1957, in a short, but influential paper. It states that if a group ''G'' has

cohomological dimension In abstract algebra, cohomological dimension is an invariant of a group which measures the homological complexity of its representations. It has important applications in geometric group theory, topology, and algebraic number theory. Cohomological ...

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. ...

K(G,1)

. For ''n'' different from 2, a group ''G'' of

''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 Mladen Bestvina (born 1959) is a Croatian-American mathematician working in the area of geometric group theory. He is a Distinguished Professor in the Department of Mathematics at the University of Utah. Biographical info Mladen Bestvina is a thr ...

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 The Whitehead conjecture (also known as the Whitehead asphericity conjecture) is a claim in algebraic topology. It was formulated by J. H. C. Whitehead in 1941. It states that every connected subcomplex of a two-dimensional aspherical CW complex ...

; in other words, not both conjectures can be true.

References

* * Conjectures Theorems in algebraic topology Unsolved problems in mathematics {{topology-stub