nLab
acyclic group

Contents

Contents

Idea

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

H (BΓ,). 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

Last revised on February 15, 2017 at 14:51:57. See the history of this page for a list of all contributions to it.