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 ${\displaystyle F\to E\to B}$ 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]

Status

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