Eilenberg–Ganea conjecture explained

K(G, 1 ) . 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

• Samuel . Eilenberg . Samuel Eilenberg. Tudor . Ganea . Tudor Ganea. On the Lusternik–Schnirelmann category of abstract groups . . 2nd Ser. . 65 . 1957 . 3 . 517–518 . 10.2307/1970062 . 1970062 . 0085510 .
• Mladen . Bestvina . Mladen Bestvina. Noel . Brady . Morse theory and finiteness properties of groups . . 129 . 1997 . 3 . 445–470 . 10.1007/s002220050168 . 1997InMat.129..445B . 1465330 . 120422255 .