Halperin conjecture

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

The statement

Suppose that ${\displaystyle F\to E\to B}$ is a fibration of simply connected spaces such that ${\displaystyle F}$ is rationally elliptic and ${\displaystyle \chi (F)\neq 0}$ (i.e., ${\displaystyle F}$ has non-zero Euler characteristic), then the Serre spectral sequence associated to the fibration collapses at the ${\displaystyle E_{2}}$ page.[1]

The status of the conjecture

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

Notes

1. Berglund
2. Lupton

References

• Berglund, Alexander (2012), Rational homotopy theory (PDF)
• Félix, Yves; Halperin, Stephen; Thomas, Jean-Claude (1993), "Elliptic spaces II", L'Enseignement Mathématique, doi:10.5169/seals-60412, MR 1225255
• 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, MR 1802847
• 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, MR 3379890
• Félix, Yves; Oprea, John; Tanré, Daniel (2008), Algebraic Models in Geometry, Oxford: Oxford University Press, ISBN 978-0-19-920651-3, MR 2403898
• Griffiths, Phillip A.; Morgan, John W. (1981), Rational Homotopy Theory and Differential Forms, Boston: Birkhäuser, ISBN 3-7643-3041-4, MR 0641551
• Hess, Kathryn (1999), "A history of rational homotopy theory", in James, Ioan M. (ed.), History of Topology, Amsterdam: North-Holland, pp. 757–796, doi:10.1016/B978-044482375-5/50028-6, ISBN 0-444-82375-1, MR 1721122
• Hess, Kathryn (2007), "Rational homotopy theory: a brief introduction" (PDF), Interactions between Homotopy Theory and Algebra, Contemporary Mathematics, 436, American Mathematical Society, pp. 175–202, arXiv:math/0604626, doi:10.1090/conm/436/08409, ISBN 9780821838143, MR 2355774
• Lupton, Gregory (1997), "Variations on a conjecture of Halperin", Homotopy and Geometry (Warsaw, 1997), arXiv:math/0010124, MR 1679854