Misplaced Pages

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 ${\displaystyle F\to E\to B}$ 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.

Status

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

Notes

1. Berglund, Alexander (2012), Rational homotopy theory (PDF)
2. Lupton, Gregory (1997), "Variations on a conjecture of Halperin", Homotopy and Geometry (Warsaw, 1997), arXiv:math/0010124, MR 1679854

Further reading

• 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, vol. 436, American Mathematical Society, pp. 175–202, arXiv:math/0604626, doi:10.1090/conm/436/08409, ISBN 9780821838143, MR 2355774

This topology-related article is a stub. You can help Misplaced Pages by expanding it.

• v
• t
• e

• Homotopy theory
• Spectral sequences
• Conjectures
• Topology stubs