Halperin conjecture
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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 → E → B {\displaystyle F\to E\to B} is a fibration of simply connected spaces such that F {\displaystyle F} is rationally elliptic and χ ( F ) ≠ 0 {\displaystyle \chi (F)\neq 0} (i.e., F {\displaystyle F} has non-zero Euler characteristic), then the Serre spectral sequence associated to the fibration collapses at the E 2 {\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
Further reading
- Félix, Yves; Halperin, Stephen; Thomas, Jean-Claude (1993), "Elliptic spaces II", L'Enseignement Mathématique, 39 (1–2): 25, doi:, MR
- Félix, Yves; Halperin, Stephen; Thomas, Jean-Claude (2001), Rational Homotopy Theory, New York: Springer Nature, doi:, ISBN 0-387-95068-0, MR
- Félix, Yves; Halperin, Stephen; Thomas, Jean-Claude (2015), Rational Homotopy Theory II, Singapore: World Scientific, doi:, ISBN 978-981-4651-42-4, MR
- Félix, Yves; Oprea, John; Tanré, Daniel (2008), Algebraic Models in Geometry, Oxford: Oxford University Press, ISBN 978-0-19-920651-3, MR
- Griffiths, Phillip A.; Morgan, John W. (1981), Rational Homotopy Theory and Differential Forms, Boston: Birkhäuser, ISBN 3-7643-3041-4, MR
- Hess, Kathryn (1999), "A history of rational homotopy theory", in James, Ioan M. (ed.), History of Topology, Amsterdam: North-Holland, pp. 757–796, doi:, ISBN 0-444-82375-1, MR
- Hess, Kathryn (2007), (PDF), Interactions between Homotopy Theory and Algebra, Contemporary Mathematics, vol. 436, American Mathematical Society, pp. 175–202, arXiv:, doi:, ISBN 9780821838143, MR