Variations on a Conjecture of Halperin

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 E2-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 numerical invariants 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. §1 — Introduction We begin with a description of the conjecture referred to in the title. In this paper, all spaces are simply connected CW complexes and are of finite type over Q, i.e., have finite-dimensional rational homology groups. A fibration F j −→ E p −→ B is said to be totally non-cohomologous to zero (abbreviated TNCZ) if the induced homomorphism j : H(E;Q) → H(F ;Q) is onto. This is a very strong condition to place on a fibration. It is equivalent to requiring that the Serre spectral sequence (for cohomology with rational coefficients) collapse at the E2-term (cf. [McC,Th.5.9]). In this case there is an isomorphismH(E;Q) ∼= H(B;Q)⊗H(F ;Q) ofH(B;Q)modules. Thus a TNCZ fibration is somewhere between being trivial from the rational homology point of view and being trivial from the rational cohomology algebra point of view (cf. Example 1.2). In the sequel we focus on certain fibre spaces F that satisfy the following conditions: (1) H(F ;Q) is finite-dimensional. (2) π∗(F )⊗ Q is finite-dimensional. (3) The Euler characteristic of F , i.e., ∑