# Acyclic space

In mathematics, an **acyclic space** is a nonempty 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,

- [math]\displaystyle{ \tilde{H}_i(X)=0, \quad \forall i\ge -1. }[/math]

It is common to consider such a space as a nonempty space without "holes"; for example, a circle or a sphere is not acyclic but a disc
or a ball is acyclic. This condition however is weaker than asking that every closed loop in the space would bound a disc in the space, all we ask is that any closed loop—and higher dimensional analogue thereof—would bound something like a "two-dimensional surface."
The condition of acyclicity on a space *X* implies, for example, for nice spaces—say, simplicial complexes—that any continuous map of *X* to the circle or to the higher spheres is null-homotopic.

If a space *X* is contractible, then it is also acyclic, by the homotopy invariance of homology. The converse is not true, in general. Nevertheless, 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 instance, 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 [math]\displaystyle{ \pi_1(X) }[/math] need not be trivial. For example, the punctured PoincarĂ© homology sphere is an acyclic, 3-dimensional manifold which is not contractible.

This gives a repertoire of examples, since the first homology group is the abelianization 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) homology groups vanish, i.e., [math]\displaystyle{ \tilde{H}_i(G;\mathbf{Z})=0 }[/math], for all [math]\displaystyle{ i\ge 0 }[/math]. Every acyclic group is thus a perfect group, meaning its first homology group vanishes: [math]\displaystyle{ H_1(G;\mathbf{Z})=0 }[/math], and in fact, a superperfect group, meaning the first two homology groups vanish: [math]\displaystyle{ H_1(G;\mathbf{Z})=H_2(G;\mathbf{Z})=0 }[/math]. The converse is not true: the binary icosahedral group is superperfect (hence perfect) but not acyclic.

## See also

## References

- Dror, Emmanuel (1972), "Acyclic spaces",
*Topology***11**: 339–348, doi:10.1016/0040-9383(72)90030-4 - Dror, Emmanuel (1973), "Homology spheres",
*Israel Journal of Mathematics***15**: 115–129, doi:10.1007/BF02764597 - Berrick, A. Jon; Hillman, Jonathan A. (2003), "Perfect and acyclic subgroups of finitely presentable groups",
*Journal of the London Mathematical Society***68**(3): 683–698, doi:10.1112/S0024610703004587

## External links

- Hazewinkel, Michiel, ed. (2001), "Acyclic groups",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/a110270

Original source: https://en.wikipedia.org/wiki/Acyclic space.
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