Acyclic space

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an acyclic space is a nonempty <a href="/facts/Topological_space/v2ut7CbK">topological space</a> X in which cycles are always boundaries, in the sense of <a href="/facts/Homology_(mathematics)/VFJmryEW">homology theory</a>. 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 <a href="/facts/Reduced_homology/ySoPPB63">reduced homology</a>,

H
                ~
              
            
          
          
            i
          
        
        (
        X
        )
        =
        0
        ,
        
        ∀
        i
        ≥
        −
        1.
      
    
    {\displaystyle {\tilde {H}}_{i}(X)=0,\quad \forall i\geq -1.}

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, <a href="/facts/Simplicial_complex/fF2282CF">simplicial complexes</a>—that any continuous map of X to the circle or to the higher spheres is <a href="/facts/Null-homotopic/1nWrPxdj">null-homotopic</a>.
If a space X is <a href="/facts/Contractible_space/REXYj8uS">contractible</a>, then it is also acyclic, by the homotopy invariance of homology. The converse is not true, in general. Nevertheless, if X is an acyclic <a href="/facts/CW_complex/rN7buLMo">CW complex</a>, and if the <a href="/facts/Fundamental_group/VyYtLqSP">fundamental group</a> of X is trivial, then X is a <a href="/facts/Contractible_space/REXYj8uS">contractible space</a>, as follows from the <a href="/facts/Whitehead_theorem/43UKropG">Whitehead theorem</a> and the <a href="/facts/Hurewicz_theorem/rHClbqTd">Hurewicz theorem</a>.

Acyclic space open-in-new

Acyclic space