Excision theorem

<p>In <a href="/facts/Algebraic_topology/2wA8yrjT">algebraic topology</a>, a branch of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the excision theorem is a theorem about <a href="/facts/Relative_homology/K3HAWAF8">relative homology</a> and one of the <a href="/facts/Eilenberg%E2%80%93Steenrod_axioms/2c42f4He">Eilenberg–Steenrod axioms</a>. Given a topological space 
  
    
      
        X
      
    
    {\displaystyle X}
  
 and subspaces 
  
    
      
        A
      
    
    {\displaystyle A}
  
 and 
  
    
      
        U
      
    
    {\displaystyle U}
  
 such that 
  
    
      
        U
      
    
    {\displaystyle U}
  
 is also a subspace of 
  
    
      
        A
      
    
    {\displaystyle A}
  
, the theorem says that under certain circumstances, we can cut out (excise) 
  
    
      
        U
      
    
    {\displaystyle U}
  
 from both spaces such that the <a href="/facts/Relative_homology/K3HAWAF8">relative homologies</a> of the pairs 
  
    
      
        (
        X
        ∖
        U
        ,
        A
        ∖
        U
        )
      
    
    {\displaystyle (X\setminus U,A\setminus U)}
  
 into 
  
    
      
        (
        X
        ,
        A
        )
      
    
    {\displaystyle (X,A)}
  
 are isomorphic.
</p><p>This assists in computation of <a href="/facts/Singular_homology/6c2AHr85">singular homology</a> groups, as sometimes after excising an appropriately chosen subspace we obtain something easier to compute.
</p>

Excision theorem open-in-new

Excision theorem