# Acyclic space

In mathematics, an acyclic space is a topological space X in which cycles are always boundaries, in the sense of homology theory. This implies that integral homology groups in all dimensions of X are isomorphic to the corresponding homology groups of a point.

In other words, using the idea of reduced homology,

$\tilde{H}_i(X)=0, \quad \forall i\ge 0.$

If X is an acyclic CW complex, and if the fundamental group of X is trivial, then X is a contractible space, as follows from the Whitehead theorem and the Hurewicz theorem.

## Examples

Acyclic spaces occur in topology, where they can be used to construct other, more interesting topological spaces.

For example, if one removes a single point from a manifold M which is a homology sphere, one gets such a space. The homotopy groups of an acyclic space X do not vanish in general, because the fundamental group need not be trivial. For example, the punctured Poincaré sphere is an acyclic, 3-dimensional manifold which is not contractible.

This gives a repertoire of examples, since the first homology group is the abelianisation of the fundamental group. With every perfect group G one can associate a (canonical, terminal) acyclic space, whose fundamental group is a central extension of the given group G.

The homotopy groups of these associated acyclic spaces are closely related to Quillen's plus construction on the classifying space BG.

## Acyclic groups

An acyclic group is a group G whose classifying space BG is acyclic; in other words, all its (reduced) group homology groups vanish ($\tilde H_i(G;\mathbf{Z})=0$). Every acyclic group is thus a perfect group (meaning first homology group vanishes: $H_1(G;\mathbf{Z})=0$), and in fact, a superperfect group (meaning first two homology groups vanish: $H_1(G;\mathbf{Z})=H_2(G;\mathbf{Z})=0$). The converse is not true: the binary icosahedral group is superperfect (hence perfect) but not acyclic.

## References

• Emmanuel Dror, "Acyclic spaces", Topology 11 (1972), 339–348. MR 0315713
• Emmanuel Dror, "Homology spheres", Israel Journal of Mathematics 15 (1973), 115–129. MR 0328926
• A. Jon Berrick and Jonathan A. Hillman, "Perfect and acyclic subgroups of finitely presentable groups", Journal of the London Mathematical Society (2) 68 (2003), no. 3, 683–698. MR 2009444