Universal coefficient theorem

In <a href="/facts/Algebraic_topology/2wA8yrjT">algebraic topology</a>, universal coefficient theorems establish relationships between <a href="/facts/Homology_group/VFJmryEW">homology groups</a> (or <a href="/facts/Cohomology_group/J7FJzTNh">cohomology groups</a>) with different coefficients. For instance, for every <a href="/facts/Topological_space/v2ut7CbK">topological space</a> X, its integral homology groups:

H
          
            i
          
        
        (
        X
        ,
        
          Z
        
        )
      
    
    {\displaystyle H_{i}(X,\mathbb {Z} )}

completely determine its homology groups with coefficients in A, for any <a href="/facts/Abelian_group/FfWwmx4R">abelian group</a> A:

H
          
            i
          
        
        (
        X
        ,
        A
        )
      
    
    {\displaystyle H_{i}(X,A)}

Here 
 
 
 
 
 H
 
 i
 
 
 
 
 {\displaystyle H_{i}}
 
 might be the <a href="/facts/Simplicial_homology/Tc3I3IJh">simplicial homology</a>, or more generally the <a href="/facts/Singular_homology/6c2AHr85">singular homology</a>. The usual proof of this result is a pure piece of <a href="/facts/Homological_algebra/Jfm8w52l">homological algebra</a> about <a href="/facts/Chain_complex/YAlchtCT">chain complexes</a> of <a href="/facts/Free_abelian_group/oaSRGmpY">free abelian groups</a>. The form of the result is that other coefficients A may be used, at the cost of using a <a href="/facts/Tor_functor/oFn2L3gd">Tor functor</a>.
For example, it is common to take 
 
 
 
 A
 
 
 {\displaystyle A}
 
 to be 
 
 
 
 
 Z
 
 
 /
 
 2
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} /2\mathbb {Z} }
 
, so that coefficients are modulo 2. This becomes straightforward in the absence of 2-<a href="/facts/Torsion_(algebra)/uNRnwtLz">torsion</a> in the homology. Quite generally, the result indicates the relationship that holds between the <a href="/facts/Betti_number/r42Y8k6f">Betti numbers</a> 
 
 
 
 
 b
 
 i
 
 
 
 
 {\displaystyle b_{i}}
 
 of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and the Betti numbers 
 
 
 
 
 b
 
 i
 ,
 F
 
 
 
 
 {\displaystyle b_{i,F}}
 
 with coefficients in a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> 
 
 
 
 F
 
 
 {\displaystyle F}
 
. These can differ, but only when the <a href="/facts/Characteristic_(algebra)/D2VzcQaG">characteristic</a> of 
 
 
 
 F
 
 
 {\displaystyle F}
 
 is a <a href="/facts/Prime_number/L20FLkgn">prime number</a> 
 
 
 
 p
 
 
 {\displaystyle p}
 
 for which there is some 
 
 
 
 p
 
 
 {\displaystyle p}
 
-torsion in the homology.

Universal coefficient theorem open-in-new

Universal coefficient theorem