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 $F\to E\to B$ 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]

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

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

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