Classifying space for SO(n)

<h2 id="definition">Definition</h2>
<p>There is a canonical inclusion of real oriented Grassmannians given by 
  
    
      
        
          
            
              
                Gr
                ~
              
            
          
          
            n
          
        
        (
        
          
            R
          
          
            k
          
        
        )
        ↪
        
          
            
              
                Gr
                ~
              
            
          
          
            n
          
        
        (
        
          
            R
          
          
            k
            +
            1
          
        
        )
        ,
        V
        ↦
        V
        ×
        {
        0
        }
      
    
    {\displaystyle {\widetilde {\operatorname {Gr} }}_{n}(\mathbb {R} ^{k})\hookrightarrow {\widetilde {\operatorname {Gr} }}_{n}(\mathbb {R} ^{k+1}),V\mapsto V\times \{0\}}
  
. Its colimit is:<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p>

BSO
        ⁡
        (
        n
        )
        :=
        
          
            
              
                Gr
                ~
              
            
          
          
            n
          
        
        (
        
          
            R
          
          
            ∞
          
        
        )
        :=
        
          lim
          
            k
            →
            ∞
          
        
        
          
            
              
                Gr
                ~
              
            
          
          
            n
          
        
        (
        
          
            R
          
          
            k
          
        
        )
        .
      
    
    {\displaystyle \operatorname {BSO} (n):={\widetilde {\operatorname {Gr} }}_{n}(\mathbb {R} ^{\infty }):=\lim _{k\rightarrow \infty }{\widetilde {\operatorname {Gr} }}_{n}(\mathbb {R} ^{k}).}

<p>Since real oriented Grassmannians can be expressed as a <a href="/facts/Homogeneous_space/YGebqcJA">homogeneous space</a> by:
</p>

Gr
                ~
              
            
          
          
            n
          
        
        (
        
          
            R
          
          
            k
          
        
        )
        =
        SO
        ⁡
        (
        n
        +
        k
        )
        
          /
        
        (
        SO
        ⁡
        (
        n
        )
        ×
        SO
        ⁡
        (
        k
        )
        )
      
    
    {\displaystyle {\widetilde {\operatorname {Gr} }}_{n}(\mathbb {R} ^{k})=\operatorname {SO} (n+k)/(\operatorname {SO} (n)\times \operatorname {SO} (k))}

<p>the group structure carries over to 
  
    
      
        BSO
        ⁡
        (
        n
        )
      
    
    {\displaystyle \operatorname {BSO} (n)}
  
.
</p>
<h2 id="simplest-classifying-spaces">Simplest classifying spaces</h2>
<ul><li>Since 
  
    
      
        SO
        ⁡
        (
        1
        )
        ≅
        1
      
    
    {\displaystyle \operatorname {SO} (1)\cong 1}
  
 is the <a href="/facts/Trivial_group/XPjQkQFm">trivial group</a>, 
  
    
      
        BSO
        ⁡
        (
        1
        )
        ≅
        {
        ∗
        }
      
    
    {\displaystyle \operatorname {BSO} (1)\cong \{*\}}
  
 is the trivial topological space.</li></ul>
<ul><li>Since 
  
    
      
        SO
        ⁡
        (
        2
        )
        ≅
        U
        ⁡
        (
        1
        )
      
    
    {\displaystyle \operatorname {SO} (2)\cong \operatorname {U} (1)}
  
, one has 
  
    
      
        BSO
        ⁡
        (
        2
        )
        ≅
        BU
        ⁡
        (
        1
        )
        ≅
        
          C
        
        
          P
          
            ∞
          
        
      
    
    {\displaystyle \operatorname {BSO} (2)\cong \operatorname {BU} (1)\cong \mathbb {C} P^{\infty }}
  
.</li></ul>
<h2 id="classification-of-principal-bundles">Classification of principal bundles</h2>
<p>Given a <a href="/facts/Topological_space/v2ut7CbK">topological space</a> 
  
    
      
        X
      
    
    {\displaystyle X}
  
 the set of 
  
    
      
        SO
        ⁡
        (
        n
        )
      
    
    {\displaystyle \operatorname {SO} (n)}
  
 principal bundles on it up to isomorphism is denoted 
  
    
      
        
          Prin
          
            SO
            ⁡
            (
            n
            )
          
        
        ⁡
        (
        X
        )
      
    
    {\displaystyle \operatorname {Prin} _{\operatorname {SO} (n)}(X)}
  
. If 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is a <a href="/facts/CW_complex/rN7buLMo">CW complex</a>, then the map:<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p>

[
        X
        ,
        BSO
        ⁡
        (
        n
        )
        ]
        →
        
          Prin
          
            SO
            ⁡
            (
            n
            )
          
        
        ⁡
        (
        X
        )
        ,
        [
        f
        ]
        ↦
        
          f
          
            ∗
          
        
        ESO
        ⁡
        (
        n
        )
      
    
    {\displaystyle [X,\operatorname {BSO} (n)]\rightarrow \operatorname {Prin} _{\operatorname {SO} (n)}(X),[f]\mapsto f^{*}\operatorname {ESO} (n)}

<p>is <a href="/facts/Bijection/ayC7YE3z">bijective</a>.
</p>
<h2 id="cohomology-ring">Cohomology ring</h2>
<p>The <a href="/facts/Cohomology_ring/eyUR2o1d">cohomology ring</a> of 
  
    
      
        BSO
        ⁡
        (
        n
        )
      
    
    {\displaystyle \operatorname {BSO} (n)}
  
 with coefficients in the <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> 
  
    
      
        
          
            Z
          
          
            2
          
        
      
    
    {\displaystyle \mathbb {Z} _{2}}
  
 of <a href="/facts/GF(2)/KIKKpYua">two elements</a> is generated by the <a href="/facts/Stiefel%25E2%2580%2593Whitney_class/H3f972We">Stiefel–Whitney classes</a>:<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>

H
          
            ∗
          
        
        (
        BSO
        ⁡
        (
        n
        )
        ;
        
          
            Z
          
          
            2
          
        
        )
        =
        
          
            Z
          
          
            2
          
        
        [
        
          w
          
            2
          
        
        ,
        …
        ,
        
          w
          
            n
          
        
        ]
        .
      
    
    {\displaystyle H^{*}(\operatorname {BSO} (n);\mathbb {Z} _{2})=\mathbb {Z} _{2}[w_{2},\ldots ,w_{n}].}

<p>The results holds more generally for every ring with <a href="/facts/Characteristic_(algebra)/D2VzcQaG">characteristic</a> 
  
    
      
        char
        =
        2
      
    
    {\displaystyle \operatorname {char} =2}
  
.
</p><p>The cohomology ring of 
  
    
      
        BSO
        ⁡
        (
        n
        )
      
    
    {\displaystyle \operatorname {BSO} (n)}
  
 with coefficients in the field 
  
    
      
        
          Q
        
      
    
    {\displaystyle \mathbb {Q} }
  
 of <a href="/facts/Rational_number/VJQmyY05">rational numbers</a> is generated by <a href="/facts/Pontryagin_class/TIFceC5j">Pontrjagin classes</a> and <a href="/facts/Euler_class/UnjlBrKe">Euler class</a>:
</p>

H
          
            ∗
          
        
        (
        BSO
        ⁡
        (
        2
        n
        )
        ;
        
          Q
        
        )
        ≅
        
          Q
        
        [
        
          p
          
            1
          
        
        ,
        …
        ,
        
          p
          
            n
          
        
        ,
        e
        ]
        
          /
        
        (
        
          p
          
            n
          
        
        −
        
          e
          
            2
          
        
        )
        ,
      
    
    {\displaystyle H^{*}(\operatorname {BSO} (2n);\mathbb {Q} )\cong \mathbb {Q} [p_{1},\ldots ,p_{n},e]/(p_{n}-e^{2}),}

H
          
            ∗
          
        
        (
        BSO
        ⁡
        (
        2
        n
        +
        1
        )
        ;
        
          Q
        
        )
        ≅
        
          Q
        
        [
        
          p
          
            1
          
        
        ,
        …
        ,
        
          p
          
            n
          
        
        ]
        .
      
    
    {\displaystyle H^{*}(\operatorname {BSO} (2n+1);\mathbb {Q} )\cong \mathbb {Q} [p_{1},\ldots ,p_{n}].}

<h2 id="infinite-classifying-space">Infinite classifying space</h2>
<p>The canonical inclusions 
  
    
      
        SO
        ⁡
        (
        n
        )
        ↪
        SO
        ⁡
        (
        n
        +
        1
        )
      
    
    {\displaystyle \operatorname {SO} (n)\hookrightarrow \operatorname {SO} (n+1)}
  
 induce canonical inclusions 
  
    
      
        BSO
        ⁡
        (
        n
        )
        ↪
        BSO
        ⁡
        (
        n
        +
        1
        )
      
    
    {\displaystyle \operatorname {BSO} (n)\hookrightarrow \operatorname {BSO} (n+1)}
  
 on their respective classifying spaces. Their respective colimits are denoted as:
</p>

SO
        :=
        
          lim
          
            n
            →
            ∞
          
        
        SO
        ⁡
        (
        n
        )
        ;
      
    
    {\displaystyle \operatorname {SO} :=\lim _{n\rightarrow \infty }\operatorname {SO} (n);}

BSO
        :=
        
          lim
          
            n
            →
            ∞
          
        
        BSO
        ⁡
        (
        n
        )
        .
      
    
    {\displaystyle \operatorname {BSO} :=\lim _{n\rightarrow \infty }\operatorname {BSO} (n).}

<p>
  
    
      
        BSO
      
    
    {\displaystyle \operatorname {BSO} }
  
 is indeed the classifying space of 
  
    
      
        SO
      
    
    {\displaystyle \operatorname {SO} }
  
.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Classifying_space_for_O(n)/Y8zCs20t">Classifying space for O(n)</a></li>
<li><a href="/facts/Classifying_space_for_U(n)/QvXR0kNz">Classifying space for U(n)</a></li>
<li><a href="/facts/Classifying_space_for_SU(n)/dwk9hqrw">Classifying space for SU(n)</a></li></ul>
<h2 id="literature">Literature</h2>
<ul><li><a href="/facts/John_Milnor/LTf24paN">Milnor, John</a>; <a href="/facts/James_Stasheff/kceG3XPX">Stasheff, James</a> (1974). <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/milnstas.pdf"><i>Characteristic classes</i></a> (PDF). Princeton University Press. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1515%2F9781400881826">10.1515/9781400881826</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780691081229.</li>
<li>Hatcher, Allen (2002). <a href="https://pi.math.cornell.edu/~hatcher/AT/ATpage.html"><i>Algebraic topology</i></a>. Cambridge: <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-79160-X.</li>
<li>Mitchell, Stephen (August 2001). <a href="https://math.mit.edu/~mbehrens/18.906/prin.pdf"><i>Universal principal bundles and classifying spaces</i></a> (PDF).{{cite book}}:  CS1 maint: year (link)</li></ul>
<h2 id="external-links">External links</h2>
<ul><li>classifying space on nLab</li>
<li>BSO(n) on nLab</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Milnor & Stasheff 74, section 12.2 The Oriented Universal Bundle on page 151 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>"universal principal bundle". nLab. Retrieved 2024-03-14. <a href="https://ncatlab.org/nlab/show/universal+principal+bundle" target="_blank">https://ncatlab.org/nlab/show/universal+principal+bundle</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Milnor & Stasheff, Theorem 12.4. <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Hatcher 02, Example 4D.6. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

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

Classifying space for SO(n)