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
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.
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
Last revised on February 15, 2017 at 19:51:57. See the history of this page for a list of all contributions to it.