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.

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