Acyclic spaces

Acyclic spaces are, by definition, topological spaces whose integral homology groups in all dimensions are isomorphic to the corresponding homology groups of the single point space, in particular H_i(X)=0, for i>0. For example, if one removes a single point from a manifold which is a homology sphere one gets such a space. The homotopy groups of an acyclic space do not vanish in general, unless the fundamental group does (and the space is a cell complex.) Acyclic spaces occure in topology, a branch of mathematics, and there they can be used to construct intersting topological spaces and in particular homology spheres. With every perfect group G one can associate an (canonical, terminal) acyclic space whose fundamental group is a central extension of the given group G. The homotpy groups of these associated acyclic spaces are closely related to Quillen's plus construction on the classifying space BG.