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.[1]
Status
As of 2019, Halperin's conjecture is still open. Gregory Lupton has reformulated the conjecture in terms of formality relations.[2]
Notes
- ^ Berglund, Alexander (2012), "Rational homotopy theory"
- ^ 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. 39 (1–2): 25, 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
- Rational Homotopy Theory and Differential Forms, Boston: Birkhäuser, 1981, ISBN 3-7643-3041-4. MR 0641551
- 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. MR 1721122
- "Rational homotopy theory: a brief introduction", Interactions between Homotopy Theory and Algebra, Vol. 436, Contemporary Mathematics, American Mathematical Society, 2007, pp. 175–202, arXiv:math/0604626. doi:10.1090/conm/436/08409. ISBN 9780821838143. MR 2355774