Halperin conjecture

<h2 id="statement">Statement</h2>
<p>Suppose that 
  
    
      
        F
        →
        E
        →
        B
      
    
    {\displaystyle F\to E\to B}
  
 is a fibration of <a href="/facts/Simply_connected_space/xFlta63J">simply connected spaces</a> such that 
  
    
      
        F
      
    
    {\displaystyle F}
  
 is <a href="/facts/Rational_homotopy_theory/b91f4L7z">rationally elliptic</a> and 
  
    
      
        χ
        (
        F
        )
        ≠
        0
      
    
    {\displaystyle \chi (F)\neq 0}
  
 (i.e., 
  
    
      
        F
      
    
    {\displaystyle F}
  
 has non-zero <a href="/facts/Euler_characteristic/JCjzWUr2">Euler characteristic</a>), then the <a href="/facts/Serre_spectral_sequence/9kAf3sYV">Serre spectral sequence</a> associated to the fibration collapses at the 
  
    
      
        
          E
          
            2
          
        
      
    
    {\displaystyle E_{2}}
  
 page.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p>
<h2 id="status">Status</h2>
<p>As of 2019, Halperin's conjecture is still open. Gregory Lupton has reformulated the conjecture in terms of formality relations.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p>
<h2 id="notes">Notes</h2>

<h2 id="further-reading">Further reading</h2>
<ul><li>Félix, Yves; Halperin, Stephen; Thomas, Jean-Claude (1993), "Elliptic spaces II", <i><a href="/facts/L%2527Enseignement_Math%25C3%25A9matique/Mn8Bevh6">L'Enseignement Mathématique</a></i>, 39 (1–2): 25, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.5169%2Fseals-60412">10.5169/seals-60412</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1225255">1225255</a></li>
<li>Félix, Yves; <a href="/facts/Stephen_Halperin/2VySVfGV">Halperin, Stephen</a>; Thomas, Jean-Claude (2001), <i>Rational Homotopy Theory</i>, New York: <a href="/facts/Springer_Nature/TpTgApAh">Springer Nature</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-1-4613-0105-9">10.1007/978-1-4613-0105-9</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-95068-0, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1802847">1802847</a></li>
<li>Félix, Yves; <a href="/facts/Stephen_Halperin/2VySVfGV">Halperin, Stephen</a>; Thomas, Jean-Claude (2015), <i>Rational Homotopy Theory II</i>, Singapore: <a href="/facts/World_Scientific/qJPTP0Yk">World Scientific</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1142%2F9473">10.1142/9473</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-981-4651-42-4, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3379890">3379890</a></li>
<li>Félix, Yves; Oprea, John; Tanré, Daniel (2008), <i>Algebraic Models in Geometry</i>, Oxford: <a href="/facts/Oxford_University_Press/CYyTzzWG">Oxford University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-19-920651-3, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2403898">2403898</a></li>
<li><a href="/facts/Phillip_Griffiths/EHQJfBVI">Griffiths, Phillip A.</a>; <a href="/facts/John_Morgan_(mathematician)/CHqR1naz">Morgan, John W.</a> (1981), <i>Rational Homotopy Theory and Differential Forms</i>, Boston: Birkhäuser, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-7643-3041-4, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0641551">0641551</a></li>
<li><a href="/facts/Kathryn_Hess/IDKJqEGL">Hess, Kathryn</a> (1999), "A history of rational homotopy theory", in <a href="/facts/Ioan_James/T7UdMHUF">James, Ioan M.</a> (ed.), <i>History of Topology</i>, Amsterdam: North-Holland, pp. 757–796, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2FB978-044482375-5%2F50028-6">10.1016/B978-044482375-5/50028-6</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-444-82375-1, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1721122">1721122</a></li>
<li><a href="/facts/Kathryn_Hess/IDKJqEGL">Hess, Kathryn</a> (2007), <a href="http://www.math.uic.edu/~bshipley/hess_ratlhtpy.pdf">"Rational homotopy theory: a brief introduction"</a> (PDF), <i>Interactions between Homotopy Theory and Algebra</i>, Contemporary Mathematics, vol. 436, <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>, pp. 175–202, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0604626">math/0604626</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2Fconm%2F436%2F08409">10.1090/conm/436/08409</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780821838143, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2355774">2355774</a></li></ul>

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

<ol>
<li id="fn:1"><p>Berglund, Alexander (2012), Rational homotopy theory (PDF) <a href="http://staff.math.su.se/alexb/rathom2.pdf" target="_blank">http://staff.math.su.se/alexb/rathom2.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Lupton, Gregory (1997), "Variations on a conjecture of Halperin", Homotopy and Geometry (Warsaw, 1997), arXiv:math/0010124, MR 1679854 <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Halperin conjecture open-in-new

Halperin conjecture