Jorge Martín: The Halperin conjecture

Abstract

The Halperin conjecture is a long-standing open problem in algebraic topology, stating that the rational Serre spectral sequence of orientable fibrations whose fibre is a positively elliptic space degenerates at the E₂ page. In this thesis, we carry out a survey of the conjecture, studying its background, various alternative formulations and the current state of research. Concretely, we first review the basics of fibrations, spectral sequences and rational homotopy theory as a basis for the rest of the work. Then, we move on to state two re-phrasings of the conjecture and show their equivalence with the original one. Our main contribution is to give complete proofs of these equivalences, whose details had been partially omitted in the literature. Especially relevant is the algebraic formulation, which states that the cohomology algebra of a positively elliptic space does not admit non-zero derivations of negative degree. Later, we present a series of specific cases for which the conjecture has been proved and review the techniques used in the latest publications on the topic. We conclude with counterexamples showing that the statement of the conjecture is sharp.