Loop space

<h2 id="eckmannhilton-duality">Eckmann–Hilton duality</h2>
<p>The loop space is dual to the <a href="/facts/Suspension_(topology)/pIWr58dM">suspension</a> of the same space; this duality is sometimes called <a href="/facts/Eckmann%25E2%2580%2593Hilton_duality/NqasGk7N">Eckmann–Hilton duality</a>. The basic observation is that
</p>

[
        Σ
        Z
        ,
        X
        ]
        ≊
        [
        Z
        ,
        Ω
        X
        ]
      
    
    {\displaystyle [\Sigma Z,X]\approxeq [Z,\Omega X]}

<p>where 
  
    
      
        [
        A
        ,
        B
        ]
      
    
    {\displaystyle [A,B]}
  
 is the set of homotopy classes of maps 
  
    
      
        A
        →
        B
      
    
    {\displaystyle A\rightarrow B}
  
,
and 
  
    
      
        Σ
        A
      
    
    {\displaystyle \Sigma A}
  
 is the suspension of A, and 
  
    
      
        ≊
      
    
    {\displaystyle \approxeq }
  
 denotes the <a href="/facts/Natural_transformation/xwPumTHt">natural</a> <a href="/facts/Homeomorphism/jMfWg3Mn">homeomorphism</a>.  This homeomorphism is essentially that of <a href="/facts/Currying/AmD3PP5B">currying</a>, modulo the quotients needed to convert the products to reduced products.
</p><p>In general, 
  
    
      
        [
        A
        ,
        B
        ]
      
    
    {\displaystyle [A,B]}
  
 does not have a group structure for arbitrary spaces 
  
    
      
        A
      
    
    {\displaystyle A}
  
 and 
  
    
      
        B
      
    
    {\displaystyle B}
  
. However, it can be shown that 
  
    
      
        [
        Σ
        Z
        ,
        X
        ]
      
    
    {\displaystyle [\Sigma Z,X]}
  
 and 
  
    
      
        [
        Z
        ,
        Ω
        X
        ]
      
    
    {\displaystyle [Z,\Omega X]}
  
 do have natural group structures when 
  
    
      
        Z
      
    
    {\displaystyle Z}
  
 and 
  
    
      
        X
      
    
    {\displaystyle X}
  
 are <a href="/facts/Pointed_space/V4DTgRCd">pointed</a>, and the aforementioned isomorphism is of those groups.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>  Thus, setting 
  
    
      
        Z
        =
        
          S
          
            k
            −
            1
          
        
      
    
    {\displaystyle Z=S^{k-1}}
  
 (the 
  
    
      
        k
        −
        1
      
    
    {\displaystyle k-1}
  
 sphere) gives the relationship
</p>

π
          
            k
          
        
        (
        X
        )
        ≊
        
          π
          
            k
            −
            1
          
        
        (
        Ω
        X
        )
      
    
    {\displaystyle \pi _{k}(X)\approxeq \pi _{k-1}(\Omega X)}
  
.
<p>This follows since the <a href="/facts/Homotopy_group/xjogL95Y">homotopy group</a> is defined as 
  
    
      
        
          π
          
            k
          
        
        (
        X
        )
        =
        [
        
          S
          
            k
          
        
        ,
        X
        ]
      
    
    {\displaystyle \pi _{k}(X)=[S^{k},X]}
  
 and the spheres can be obtained via suspensions of each-other, i.e. 
  
    
      
        
          S
          
            k
          
        
        =
        Σ
        
          S
          
            k
            −
            1
          
        
      
    
    {\displaystyle S^{k}=\Sigma S^{k-1}}
  
.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Bott_periodicity/GqgJRllU">Bott periodicity</a></li>
<li><a href="/facts/Eilenberg%25E2%2580%2593MacLane_space/mrux6oMJ">Eilenberg–MacLane space</a></li>
<li><a href="/facts/Free_loop/hDDzS8dt">Free loop</a></li>
<li><a href="/facts/Fundamental_group/VyYtLqSP">Fundamental group</a></li>
<li><a href="/facts/Gray%2527s_conjecture/aKGCgvlx">Gray's conjecture</a></li>
<li><a href="/facts/List_of_topologies/mb4VoHSD">List of topologies</a></li>
<li><a href="/facts/Loop_group/xTvCs9rM">Loop group</a></li>
<li><a href="/facts/Path_(topology)/MkKRCYXT">Path (topology)</a></li>
<li><a href="/facts/Quasigroup/LRWCA2oB">Quasigroup</a></li>
<li><a href="/facts/Spectrum_(topology)/mWho3roy">Spectrum (topology)</a></li>
<li><a href="/facts/Path_space_(algebraic_topology)/XeS9AWD2">Path space (algebraic topology)</a></li></ul>

<ul><li><a href="/facts/Frank_Adams/fxgpN9Xk">Adams, John Frank</a> (1978), <a href="https://books.google.com/books?id=e2rYkg9lGnsC"><i>Infinite loop spaces</i></a>, Annals of Mathematics Studies, vol. 90, <a href="/facts/Princeton_University_Press/kwvFlKEj">Princeton University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-691-08207-3, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0505692">0505692</a></li>
<li><a href="/facts/J._Peter_May/GPbRcPTW">May, J. Peter</a> (1972), <a href="http://www.math.uchicago.edu/~may/BOOKSMaster.html"><i>The Geometry of Iterated Loop Spaces</i></a>, Lecture Notes in Mathematics, vol. 271, Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBFb0067491">10.1007/BFb0067491</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-05904-2, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0420610">0420610</a></li></ul>

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

<ol>
<li id="fn:1"><p>May, J. P. (1999), A Concise Course in Algebraic Topology (PDF), U. Chicago Press, Chicago, retrieved 2016-08-27 (See chapter 8, section 2) <a href="/wiki/J._Peter_May" target="_blank">/wiki/J._Peter_May</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Topospaces wiki – Loop space of a based topological space <a href="http://topospaces.subwiki.org/wiki/Loop_space_of_a_based_topological_space" target="_blank">http://topospaces.subwiki.org/wiki/Loop_space_of_a_based_topological_space</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Loop space open-in-new

Loop space