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Halperin conjecture

{{short description|Mathematical conjecture}}

In rational homotopy theory, the Halperin conjecture concerns the Serre spectral sequence of certain fibrations. It is named after the Canadian mathematician Stephen Halperin.

Statement

Suppose that F \to E \to B is a fibration of simply connected spaces such that F is rationally elliptic and \chi(F) \neq 0 (i.e., F has non-zero Euler characteristic), then the Serre spectral sequence associated to the fibration collapses at the E_2 page.{{citation | last=Berglund | first=Alexander | title=Rational homotopy theory| year=2012 | url = http://staff.math.su.se/alexb/rathom2.pdf}}

Status

As of 2019, Halperin's conjecture is still open. Gregory Lupton has reformulated the conjecture in terms of formality relations.{{citation | last=Lupton | first=Gregory | title=Variations on a conjecture of Halperin | journal= Homotopy and Geometry (Warsaw, 1997)| year=1997 | arxiv=math/0010124 | mr=1679854 }}

Notes

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Further reading

  • {{citation | last1=Félix | first1=Yves | last2=Halperin | first2=Stephen | last3=Thomas | first3=Jean-Claude | title=Elliptic spaces II | journal=L'Enseignement Mathématique | year=1993 | volume=39 | issue=1–2 | page=25 | doi=10.5169/seals-60412 | mr=1225255}}
  • {{citation|last1= Félix|first1=Yves|last2=Halperin|first2=Stephen|author2-link=Stephen Halperin|last3=Thomas|first3=Jean-Claude|title=Rational Homotopy Theory|publisher=Springer Nature|location=New York|year=2001|isbn=0-387-95068-0|mr=1802847|doi=10.1007/978-1-4613-0105-9}}
  • {{citation|last1= Félix|first1=Yves|last2=Halperin|first2=Stephen|last3=Thomas|author2-link=Stephen Halperin|first3=Jean-Claude|title=Rational Homotopy Theory II|publisher=World Scientific|location=Singapore|year=2015|isbn=978-981-4651-42-4|mr=3379890|doi=10.1142/9473}}
  • {{citation|last1= Félix|first1=Yves|last2=Oprea|first2=John|last3=Tanré|first3=Daniel|title=Algebraic Models in Geometry|publisher=Oxford University Press|location=Oxford|year=2008|isbn=978-0-19-920651-3|mr=2403898}}
  • {{citation|author1-last=Griffiths|author1-first=Phillip A.|author1-link=Phillip Griffiths|author2-last=Morgan|author2-first=John W.|author2-link=John Morgan (mathematician) | title=Rational Homotopy Theory and Differential Forms | publisher=Birkhäuser | year=1981 | isbn=3-7643-3041-4 | location=Boston | mr=0641551}}
  • {{citation|author1-last=Hess|author1-first=Kathryn|author1-link=Kathryn Hess|chapter=A history of rational homotopy theory|title= History of Topology|pages=757–796|publisher= North-Holland |location=Amsterdam|year= 1999|editor-first=Ioan M.|editor-last= James| editor-link=Ioan James|isbn=0-444-82375-1|mr=1721122|doi=10.1016/B978-044482375-5/50028-6}}
  • {{citation | author1-last=Hess | author1-first=Kathryn | author1-link=Kathryn Hess | chapter=Rational homotopy theory: a brief introduction | title=Interactions between Homotopy Theory and Algebra | volume=436 | pages=175–202 | publisher=American Mathematical Society | year=2007 | mr=2355774 | chapter-url=http://www.math.uic.edu/~bshipley/hess_ratlhtpy.pdf | doi=10.1090/conm/436/08409| arxiv=math/0604626 | series=Contemporary Mathematics | isbn=9780821838143 }}

Category:Homotopy theory

Category:Spectral sequences

Category:Conjectures

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