nLab acyclic group

Idea

A discrete group $\Gamma$ is called an acyclic group if its classifying space $B \Gamma$ is an “acyclic space” in that its ordinary cohomology $H^\bullet(B \Gamma, \mathbb{Z})$ vanishes in all positive degree, so that

$$H^\bullet(B \Gamma, \mathbb{Z}) \simeq \mathbb{Z} \,.$$

Hence to cohomology these spaces look like contractible topological spaces. The existence of acyclic groups is one reason why rational homotopy theory restricts attention to simply-connected topological spaces.

References

The first two non-trivial examples of an acyclic group were apparently given in

• G. Baumslag, K.W. Gruenberg, Some reflections on cohomological dimension and freeness, J. Algebra 6 (1967), 394–409.

• D. B. A. Epstein, A group with zero homology, Proc. Camb. Phil. Soc. 64 (1968), 599–601

See also

• A. J. Berrick, The acyclic group dichotomy (arXiv:1006.4009)

• Wikipedia, Acyclic group