Variations on a conjecture of Halperin

Details Variations on a conjecture of Halperin Variations on a conjecture of Halperin

2000-10-12

arxiv.org/abs/math/0010124v1

Homotopy and Geometry, Banach Centre Publications vol. 45 (1998), 115--135

Mathematics

Algebraic Topology

Scientific paper

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.

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