In the subject of manifold theory in mathematics, if 
  
    
      
        M
      
    
    {\displaystyle M}
  
 is a topological manifold with boundary, its double is obtained by gluing two copies of 
  
    
      
        M
      
    
    {\displaystyle M}
  
 together along their common boundary. Precisely, the double is 
  
    
      
        M
        ×
        {
        0
        ,
        1
        }
        
          /
        
        ∼
      
    
    {\displaystyle M\times \{0,1\}/\sim }
  
 where 
  
    
      
        (
        x
        ,
        0
        )
        ∼
        (
        x
        ,
        1
        )
      
    
    {\displaystyle (x,0)\sim (x,1)}
  
 for all 
  
    
      
        x
        ∈
        ∂
        M
      
    
    {\displaystyle x\in \partial M}
  
. 
 If 
  
    
      
        M
      
    
    {\displaystyle M}
  
 has a smooth structure, then its double can be endowed with a smooth structure thanks to a collar neighbourdhood.: th. 9.29 & ex. 9.32 
 Although the concept makes sense for any manifold, and even for some non-manifold sets such as the Alexander horned sphere, the notion of double tends to be used primarily in the context that 
  
    
      
        ∂
        M
      
    
    {\displaystyle \partial M}
  
 is non-empty and 
  
    
      
        M
      
    
    {\displaystyle M}
  
 is compact.
 Given a manifold 
  
    
      
        M
      
    
    {\displaystyle M}
  
, the double of 
  
    
      
        M
      
    
    {\displaystyle M}
  
 is the boundary of 
  
    
      
        M
        ×
        [
        0
        ,
        1
        ]
      
    
    {\displaystyle M\times [0,1]}
  
.  This gives doubles a special role in cobordism.
 The n-sphere is the double of the n-ball. In this context, the two balls would be the upper and lower hemi-sphere respectively.  More generally, if 
  
    
      
        M
      
    
    {\displaystyle M}
  
 is closed, the double of 
  
    
      
        M
        ×
        
          D
          
            k
          
        
      
    
    {\displaystyle M\times D^{k}}
  
 is 
  
    
      
        M
        ×
        
          S
          
            k
          
        
      
    
    {\displaystyle M\times S^{k}}
  
. Even more generally, the double of a disc bundle over a manifold is a sphere bundle over the same manifold. More concretely, the double of the Möbius strip is the Klein bottle.
 If 
  
    
      
        M
      
    
    {\displaystyle M}
  
 is a closed, oriented manifold and if 
  
    
      
        
          M
          ′
        
      
    
    {\displaystyle M'}
  
 is obtained from 
  
    
      
        M
      
    
    {\displaystyle M}
  
 by removing an open ball, then the connected sum 
  
    
      
        M
        
          
            #
          
        
        −
        M
      
    
    {\displaystyle M{\mathrel {\#}}-M}
  
 is the double of 
  
    
      
        
          M
          ′
        
      
    
    {\displaystyle M'}
  
.   
 The double of a Mazur manifold is a homotopy 4-sphere.
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