# Acyclic space

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{{technical|date=June 2012}}

In [mathematics](/source/mathematics), an '''acyclic space''' is a nonempty [topological space](/source/topological_space) ''X'' in which cycles are always boundaries, in the sense of [homology theory](/source/Homology_(mathematics)). 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](/source/reduced_homology),

:<math>\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&mdash;and higher dimensional analogue thereof&mdash;would bound something like a "two-dimensional surface." 
The condition of acyclicity on a space ''X'' implies, for example, for nice spaces&mdash;say, [simplicial complex](/source/simplicial_complex)es&mdash;that any continuous map of ''X'' to the circle or to the higher spheres is [null-homotopic](/source/null-homotopic).

If a space ''X'' is [contractible](/source/contractible_space), 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](/source/CW_complex), and if the [fundamental group](/source/fundamental_group) of ''X'' is trivial, then ''X'' is a [contractible space](/source/contractible_space), as follows from the [Whitehead theorem](/source/Whitehead_theorem) and the [Hurewicz theorem](/source/Hurewicz_theorem).

==Examples==

Acyclic spaces occur in [topology](/source/topology), where they can be used to construct other, more interesting topological spaces.

For instance, if one removes a single point from a [manifold](/source/manifold) ''M'' which is a [homology sphere](/source/homology_sphere), one gets such a space. The [homotopy group](/source/homotopy_group)s of an acyclic space ''X'' do not vanish in general, because the fundamental group <math>\pi_1(X)</math> need not be trivial. For example, the punctured [Poincaré homology sphere](/source/Poincar%C3%A9_homology_sphere) is an acyclic, [3-dimensional manifold](/source/3-manifold) which is not contractible.

This gives a repertoire of examples, since the first homology group is the [abelianization](/source/Commutator_subgroup) of the fundamental group. With every [perfect group](/source/perfect_group) ''G'' one can associate a (canonical, terminal) acyclic space, whose fundamental group is a [central extension](/source/Group_extension) of the given group ''G''.

The homotopy groups of these associated acyclic spaces are closely related to [Quillen](/source/Daniel_Quillen)'s [plus construction](/source/plus_construction) on the [classifying space](/source/classifying_space) ''BG''.

==Acyclic groups==
An '''acyclic group''' is a group ''G'' whose [classifying space](/source/classifying_space) ''BG'' is acyclic; in other words, all its (reduced) [homology](/source/group_homology) groups vanish, i.e., <math>\tilde{H}_i(G;\mathbf{Z})=0</math>, for all <math>i\ge 0</math>. Every acyclic group is thus a [perfect group](/source/perfect_group), meaning its first homology group vanishes: <math>H_1(G;\mathbf{Z})=0</math>, and in fact, a [superperfect group](/source/superperfect_group), meaning the first two homology groups vanish: <math>H_1(G;\mathbf{Z})=H_2(G;\mathbf{Z})=0</math>. The converse is not true: the [binary icosahedral group](/source/binary_icosahedral_group) is superperfect (hence perfect) but not acyclic.

==See also==
* [Aspherical space](/source/Aspherical_space)

==References==
* {{citation
| last=Dror | first=Emmanuel
| title=Acyclic spaces
| journal=[Topology](/source/Topology_(journal))
| volume=11
| date=1972
| issue=4
| pages=339&ndash;348
| mr=0315713
| doi=10.1016/0040-9383(72)90030-4 | doi-access=}}
* {{citation
| last=Dror | first=Emmanuel
| title=Homology spheres
| journal=[Israel Journal of Mathematics](/source/Israel_Journal_of_Mathematics)
| volume=15
| date=1973
| issue=2
| pages=115&ndash;129
| mr=0328926
| doi=10.1007/BF02764597 | doi-access=}}
* {{citation
| last1=Berrick | first1=A. Jon
| last2=Hillman | first2=Jonathan A.
| title=Perfect and acyclic subgroups of finitely presentable groups
| journal=[Journal of the London Mathematical Society](/source/Journal_of_the_London_Mathematical_Society)
| volume=68
| date=2003
| issue=3
| pages=683&ndash;698
| mr=2009444
| doi=10.1112/S0024610703004587| s2cid=30232002
}}

==External links==
* {{springer|title=Acyclic groups|id=p/a110270}}

Category:Algebraic topology
Category:Homology theory
Category:Homotopy theory

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