  Halperin has conjectured that the Serre spectral sequence of any fibration
that has fibre space a certain kind of elliptic space should collapse at the
E_2-term. In this paper we obtain an equivalent phrasing of this conjecture, in
terms of formality relations between base and total spaces in such a fibration
(Theorem 3.4). Also, we obtain results on relations between various
numericalinvariants of the base, total and fibre spaces in these fibrations.
Some of our results give weak versions of Halperin's conjecture (Remark 4.4 and
Corollary 4.5). We go on to establish some of these weakened forms of the
conjecture (Theorem 4.7). In the last section, we discuss extensions of our
results and suggest some possibilities for future work.
