Halperin 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 [math]\displaystyle{ F \to E \to B }[/math] is a fibration of simply connected spaces such that [math]\displaystyle{ F }[/math] is rationally elliptic and [math]\displaystyle{ \chi(F) \neq 0 }[/math] (i.e., [math]\displaystyle{ F }[/math] has non-zero Euler characteristic), then the Serre spectral sequence associated to the fibration collapses at the [math]\displaystyle{ E_2 }[/math] page.^[1]

Status

As of 2019, Halperin's conjecture is still open. Gregory Lupton has reformulated the conjecture in terms of formality relations.^[2]

Notes

References

• Berglund, Alexander (2012), Rational homotopy theory, http://staff.math.su.se/alexb/rathom2.pdf 
• Félix, Yves; Halperin, Stephen; Thomas, Jean-Claude (1993), "Elliptic spaces II", L'Enseignement Mathématique, doi:10.5169/seals-60412 
• Félix, Yves; Halperin, Stephen; Thomas, Jean-Claude (2001), Rational Homotopy Theory, New York: Springer Nature, doi:10.1007/978-1-4613-0105-9, ISBN 0-387-95068-0 
• Félix, Yves; Halperin, Stephen; Thomas, Jean-Claude (2015), Rational Homotopy Theory II, Singapore: World Scientific, doi:10.1142/9473, ISBN 978-981-4651-42-4 
• Félix, Yves; Oprea, John; Tanré, Daniel (2008), Algebraic Models in Geometry, Oxford: Oxford University Press, ISBN 978-0-19-920651-3 
• Rational Homotopy Theory and Differential Forms, Boston: Birkhäuser, 1981, ISBN 3-7643-3041-4 
• James, Ioan M., ed. (1999), "A history of rational homotopy theory", History of Topology, Amsterdam: North-Holland, pp. 757–796, doi:10.1016/B978-044482375-5/50028-6, ISBN 0-444-82375-1 
• "Rational homotopy theory: a brief introduction", Interactions between Homotopy Theory and Algebra, Contemporary Mathematics, 436, American Mathematical Society, 2007, pp. 175–202, doi:10.1090/conm/436/08409, ISBN 9780821838143, http://www.math.uic.edu/~bshipley/hess_ratlhtpy.pdf 
• Lupton, Gregory (1997), "Variations on a conjecture of Halperin", Homotopy and Geometry (Warsaw, 1997) 

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