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.

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.

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

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