Acyclic space

<h2 id="examples">Examples</h2>
<p>Acyclic spaces occur in <a href="/facts/Topology/2EqW47cU">topology</a>, where they can be used to construct other, more interesting topological spaces.
</p><p>For instance, if one removes a single point from a <a href="/facts/Manifold/rXPERJtu">manifold</a> <i>M</i> which is a <a href="/facts/Homology_sphere/8XnrYQ1T">homology sphere</a>, one gets such a space. The <a href="/facts/Homotopy_group/xjogL95Y">homotopy groups</a> of an acyclic space <i>X</i> do not vanish in general, because the fundamental group 
  
    
      
        
          π
          
            1
          
        
        (
        X
        )
      
    
    {\displaystyle \pi _{1}(X)}
  
 need not be trivial. For example, the punctured <a href="/facts/Poincar%C3%A9_homology_sphere/8XnrYQ1T">Poincaré homology sphere</a> is an acyclic, <a href="/facts/3-manifold/48yr0Tzx">3-dimensional manifold</a> which is not contractible.
</p><p>This gives a repertoire of examples, since the first homology group is the <a href="/facts/Commutator_subgroup/oK9WphdE">abelianization</a> of the fundamental group. With every <a href="/facts/Perfect_group/rxztKKjd">perfect group</a> <i>G</i> one can associate a (canonical, terminal) acyclic space, whose fundamental group is a <a href="/facts/Group_extension/pGoe0l4d">central extension</a> of the given group <i>G</i>.
</p><p>The homotopy groups of these associated acyclic spaces are closely related to <a href="/facts/Daniel_Quillen/E4Nak15V">Quillen</a>'s <a href="/facts/Plus_construction/GdyNMtlj">plus construction</a> on the <a href="/facts/Classifying_space/gGIAmhtL">classifying space</a> <i>BG</i>.
</p>
<h2 id="acyclic-groups">Acyclic groups</h2>
<p>An acyclic group is a group <i>G</i> whose <a href="/facts/Classifying_space/gGIAmhtL">classifying space</a> <i>BG</i> is acyclic; in other words, all its (reduced) <a href="/facts/Group_homology/ncLTIHj4">homology</a> groups vanish, i.e., 
  
    
      
        
          
            
              
                H
                ~
              
            
          
          
            i
          
        
        (
        G
        ;
        
          Z
        
        )
        =
        0
      
    
    {\displaystyle {\tilde {H}}_{i}(G;\mathbf {Z} )=0}
  
, for all 
  
    
      
        i
        ≥
        0
      
    
    {\displaystyle i\geq 0}
  
. Every acyclic group is thus a <a href="/facts/Perfect_group/rxztKKjd">perfect group</a>, meaning its first homology group vanishes: 
  
    
      
        
          H
          
            1
          
        
        (
        G
        ;
        
          Z
        
        )
        =
        0
      
    
    {\displaystyle H_{1}(G;\mathbf {Z} )=0}
  
, and in fact, a <a href="/facts/Superperfect_group/kJIYgf8J">superperfect group</a>, meaning the first two homology groups vanish: 
  
    
      
        
          H
          
            1
          
        
        (
        G
        ;
        
          Z
        
        )
        =
        
          H
          
            2
          
        
        (
        G
        ;
        
          Z
        
        )
        =
        0
      
    
    {\displaystyle H_{1}(G;\mathbf {Z} )=H_{2}(G;\mathbf {Z} )=0}
  
. The converse is not true: the <a href="/facts/Binary_icosahedral_group/9JmfJ4Rc">binary icosahedral group</a> is superperfect (hence perfect) but not acyclic.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Aspherical_space/M43BNIGF">Aspherical space</a></li></ul>

<ul><li>Dror, Emmanuel (1972), "Acyclic spaces", <i><a href="/facts/Topology_(journal)/D2ejns02">Topology</a></i>, 11 (4): 339–348, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0040-9383%2872%2990030-4">10.1016/0040-9383(72)90030-4</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0315713">0315713</a></li>
<li>Dror, Emmanuel (1973), "Homology spheres", <i><a href="/facts/Israel_Journal_of_Mathematics/mSGtfzpS">Israel Journal of Mathematics</a></i>, 15 (2): 115–129, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF02764597">10.1007/BF02764597</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0328926">0328926</a></li>
<li>Berrick, A. Jon; Hillman, Jonathan A. (2003), "Perfect and acyclic subgroups of finitely presentable groups", <i><a href="/facts/Journal_of_the_London_Mathematical_Society/jdcf3vhX">Journal of the London Mathematical Society</a></i>, 68 (3): 683–698, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1112%2FS0024610703004587">10.1112/S0024610703004587</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2009444">2009444</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:30232002">30232002</a></li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="https://www.encyclopediaofmath.org/index.php?title=Acyclic_groups">"Acyclic groups"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li></ul>

Acyclic space open-in-new

Acyclic space