Classifying space for SO(n)

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the <a href="/facts/Classifying_space/gGIAmhtL">classifying space</a> 
  
    
      
        BSO
        ⁡
        (
        n
        )
      
    
    {\displaystyle \operatorname {BSO} (n)}
  
 for the <a href="https://en.wikipedia.org/?title=Special_orthogonal_group">special orthogonal group</a> 
  
    
      
        SO
        ⁡
        (
        n
        )
      
    
    {\displaystyle \operatorname {SO} (n)}
  
 is the base space of the <a href="/facts/Universal_bundle/N6zTy9fn">universal</a> 
  
    
      
        SO
        ⁡
        (
        n
        )
      
    
    {\displaystyle \operatorname {SO} (n)}
  
 <a href="/facts/Principal_bundle/bbB2D94B">principal bundle</a> 
  
    
      
        ESO
        ⁡
        (
        n
        )
        →
        BSO
        ⁡
        (
        n
        )
      
    
    {\displaystyle \operatorname {ESO} (n)\rightarrow \operatorname {BSO} (n)}
  
. This means that 
  
    
      
        SO
        ⁡
        (
        n
        )
      
    
    {\displaystyle \operatorname {SO} (n)}
  
 principal bundles over a <a href="/facts/CW_complex/rN7buLMo">CW complex</a> up to isomorphism are in bijection with <a href="/facts/Homotopy/1nWrPxdj">homotopy</a> classes of its <a href="/facts/Continuous_function/sKbl02pB">continuous</a> maps into 
  
    
      
        BSO
        ⁡
        (
        n
        )
      
    
    {\displaystyle \operatorname {BSO} (n)}
  
. The isomorphism is given by <a href="/facts/Pullback_bundle/nBdt2Cx7">pullback</a>.
</p>

Classifying space for SO(n) open-in-new

Classifying space for SO(n)