Halperin conjecture

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]

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

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• Lupton, Gregory (1997), "Variations on a conjecture of Halperin", Homotopy and Geometry (Warsaw, 1997), arXiv:math/0010124, MR 1679854